Calculus 2 Fundamentals - Indefinite Integrals (Integration - Single, Multivariable, Trigonometric) | Tanmay Varshney | Skillshare

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Calculus 2 Fundamentals - Indefinite Integrals (Integration - Single, Multivariable, Trigonometric)

teacher avatar Tanmay Varshney, Software Developer, Tech Educator

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Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more

Lessons in This Class

26 Lessons (3h 3m)
    • 1. Introduction to Indefinite Integrals

    • 2. 01 Anti derivative

    • 3. 02 Concept of Integration

    • 4. 03 Notation of Integration

    • 5. 04 Constant of Integration

    • 6. 05 Power rule of Integration

    • 7. 06 Sum Rule of Integration

    • 8. 07 Constant Coefficient Rule

    • 9. 08 Linearity in Integration

    • 10. 09 Inverse Trigonometric functions part 1

    • 11. 10 Inverse trigonometry Part 2

    • 12. 11 Integration of Common Functions

    • 13. 12 Substitution Rule

    • 14. 13 Substitution Rule - Examples

    • 15. 14 Integration of Rational Functions

    • 16. 15 Integration of Rational Functions - Example

    • 17. 16 Integration of Irrational functions

    • 18. 17 Integration of Irrational functions - Examples

    • 19. 18 Integration using trigonometric identities

    • 20. 19 Integration using Trigonometric identities - Example

    • 21. 20 Integration of Special functions - Part 1

    • 22. 21 Integration of Special functions - Part 2

    • 23. 22 Integration by parts

    • 24. 23 Integration by parts - Example

    • 25. 24 Miscellaneous Examples - Part 1

    • 26. 25 Miscellaneous Examples - Part 2

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About This Class

About this course:

This course focuses on all the fundamental concepts of Integral calculus - Indefinite Integrals.

You will crush the basics of The Indefinite Integral in this series.

After completing this course, you will be thoroughly prepared to learn the applications of integration like finding the area under the curve, calculating arc length, the surface area of complex functions, and many more.


What is Integral Calculus?

Integral Calculus or integration is all about joining smaller pieces to find the total. 

It is actually the opposite of Differential Calculus.


Practical Applications: 

Integral Calculus has numerous applications in the fields of:

  1. Physics: to calculate the center of mass, center of gravity, moment of inertia, etc

  2. Chemistry: to determine the rate of a chemical reaction

  3. Medical Sciences

  4. Modelling Electronic Circuits 

  5. Predicting Stock Market prices and many more

Hence, it is everywhere.

Course content:

This course has been designed for you to cover all the topics of the Indefinite integral in an exhaustive manner.

This course is categorized into:

  1. Video lectures covering the core concepts of Indefinite integral calculus and varied examples.

  2. Miscellaneous Examples to crush the basics even further.

  3. Practice Worksheets to let you crush the learned concepts even further.

Meet Your Teacher

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Tanmay Varshney

Software Developer, Tech Educator


I am a Senior Software Engineer with vast experience of working in top tech giant companies.
I have more than 6 years of industry and teaching experience in domains like:

1. Designing scalable architecture for complex and distributed systems.

2. Developing components in a system across the full stack.

3. Solving complex data structures and algorithms related problems.

These are the major skills needed to be a good software developer who can excel in any tech company easily. I am really passionate about sharing my knowledge and expertise with you.

Thus, I am on board to create awesome technical courses on Skillshare based on my expertise which can be understood in the simplest manner.

Come, join me in this learning adventure!! I will... See full profile

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1. Introduction to Indefinite Integrals: so guys, I didn't your belts. It's time to join me in this journey full of Latin. The first section you're going to learn the indefinite in eagle we will start by what is meant by anti derivative are integral. Then we will see the concept off in English and the notations used in indignation. We will see what is the constant off indignation, and we will learn the most important rule off indignation power. We will proceed further. To learn is some rule and a constant factor rule and apply them to learn the linearity in integration and you will be introduced to the inverse techno metric functions and you will see how they're used in in Biggles. Then you will learn in eagles off common functions in helpless. After this, it will cover the famous substitution rule. Then we will see integration off for national and international functions. We will see how pragmatic identities and be used to make integration. Easier Way will look at integration off some special functions. After this, we will learn about the savior rule integration by parts well, then practice variety off challenging examples in the miscellaneous example section and finally recover in Eagles off some more types. At the end, you will revise all the learned and steps off indefinite and eagle in the division at a glance section. So let's get started with this section. 2. 01 Anti derivative: Hi guys. We're going to kick start the scores by introducing you to the concept off and in elevators . The anti derivative of a function F is a function whose derivative is f itself. So let us right that and he did. It would do awful function if it's ah, function. Who's very victim is if it's a if you don't remember what a derivative is a very good deal is the rate off change off a function with respect to some variable on which that function is dependent upon. Now, if you want to understand anyway, does better. I highly recommend you to enroll in our helpless fundamental schools, which covers each off the topic on limits and derivatives in a very elaborate way. Coming back to enter data with you. We can also see if we can also see capital left is and didn't do it off and to deliver it all for function. Small left. If F dash is equal to smaller, the dash or prime represents the derivative. So to find the nd derivative for a function F, we can often ah, reverse the process off difference Asian who understand it. So, for example, let's say we have to Fine In eagle our integration You can see off Blix so we can start thinking off A function with different station is two X so that function is X squared. So this X square is the integral off Two weeks since the different station off X square is Twix, right? So the integral off two x is X squared. So by reversing the process off different station, we can easily adapt arses to find the integral off a function. So let us right this little bit victory victory Italy that in Diggle off two x is a function who's different? That this is the letter. This is the different station on this thing Is the indignation so different Season off X square Is this two weeks. So on the similar rounds uh, vice a versa that an eagle off this two weeks will be extra. So we just need to think off a function whose different station is that function that this two x here. So let's consider another example to make us more family with what I have stated about. So let's say we have a function, eh? Physical do three x squared. No, we have to find the antidote ever took off dysfunction. So the ended elevated off a function will be. We need to think off a function whose different season is this three. Expert So that function by hidden trial in your mind, you can think off as ethical excuse, since you can check this. If it's correct or not, let's strike the difference. See this excuse So me by the X we can see the difference. Asian off excuse with respect to X is equal Duke three Expert, which is the function given in the question. So we can easily say in Eagle Off three X squared is excused so mathematically we can write if d by d. X off a function Capital X is equal to small FX. Then in giggle ethics, I introduce this, um, generation little bit the later Aspinall. You can just think Think of it as in legal. So in Eagle off FX is capital ethics. So if the body aches of capital effects is equal to small effects than an eagle off small, if X is equal to a prolific sex. So finally, to conclude the take away from the lecture is that Andy, derivative of a function have it is nothing but a function whose derivative is, if that is kept 11 heart Argus. So this was it for this video. Thank you for watching. 3. 02 Concept of Integration: Hello, friends. In this video, we're going to dive deep into the concept of integration and understand what it actually means in delusion or any derivative. They both are the same thing. When dealing with indefinite and giggles. We often use the dumb, anti deliberate event in the integration interchangeably. But for sticking to a convention, we will use the term immigration more often in this course. Now, what is the notion off in duration? Integration is basically of a tow. Accumulate all the smaller pieces to find the whole. So this is the notion behind integration. No, in division has various applications. It can be used to find the area volume and compute sent central points and many other useful things. No, let's suppose we need to find the area under some car. Let's say this is FX off a function right, so we need to find a D'Andre dysfunction effects bounded by the XX is So how do we proceed about it so off first week, we can think about this. Uh, let's say we can compute the value of this function FX at various points like this, and for each point we can take ah slaves off with Delta X. Now, what is date X date X is a small change in the value of X. This is this still represents change. So this doesn't x means change in very affects. This is smaller. No, for every value off FX, we can take of it off the decks and some of all the slices to find the area. So that can be like lack civic. Calculate the value off the function. At the point I on at that point we multiply it by detects. So that will be the area off one slice or you can see one strip Andi needto do the submission off all the slices to find the area. So this will be summation. And now summation will be from eyes equal to zero and let's editor and strips. So that would be eyes equal to zero still, and so this will be the, um, formula for calculating the area. But you can see that and this approach if it take the slices off with Delta X and a good area, we find that the area is not calculated inaccurately so that you can see from here We are missing so much, so much off area in our in our calculation. So this is not me a great way off calculating this dot in the definitely Norton accurately off calculatingly area, so hardly optimize it further. Vic in Ah, that manner, we can optimize it by making this that takes this with off changing x a little bit more small. So in that way, you can find you can see from this figure if he make the date x smaller and then ah, veep, we will end up having more slices or you can see more strips than before so strips will be increased on in this way, you can see the area left behind is very, very small, very small as compared to the previous ah method off canceling the area. So this is more accurate towards calculating the area of the function when we increased the with off this change in X. So as the slices approach really close to zero in with the answer upto approaches the true answer. So let's say this day X is very close to zero. No, since they x approaches to zero, we can replace this Delta X by D x, so d access differential affects with indicates that the change in X is very in financing villages very close to zero. So this is the most optimal way off calculating the idiom. And we assume the change in X is very close to zero. So in that way, we can add up all the values multiplied by these strip Uh, strip with that is DX and add up all the all systems, so that will be weaken right. Begin the place. Sigma ethics I day X by in eagle So this sigma gets a place by integral and that will be so This will be if, but this will be affects only. So this will be in Diggle fx DX. So this is the notion behind the integration where we used the this notion to find the area off the function under a graph right under, like, 80 all of this function FX with the horizontal X success. So this was it for this particular video. Join me in the next video. Thank you. 4. 03 Notation of Integration: Hi guys. In this video you'll get to know the common notations using in giggles. First of all, the notation used for the indefinite individual is in Diggle FX Dicks. So this is this expression Israelis indefinite in giggle off FX with respect to X not a symbol, this symbol DX. If gordy differential off the variable X, it signifies that the in eagle or the derivative is with respect to X the function F X is all the immigrant off the indeed, All right. And this is the in Diegel. Sane and X is the variable off integration. Now, if you assume that capital FX is the integral off this ethics, then we can write in Diggle ethics. D X is equal to capital FX less see. In this context, see is called the constant of integration. Don't worry. We will discuss about this constant of integration in the upcoming video. So this is the constant of integration, and we will be discussing in the upcoming video about this constant off integration. So this was it for this video. Thanks for watching 5. 04 Constant of Integration: What's up, guys? In this lecture, we're going to understand what is the average three constant off integration and why it is added in. The result often giggles. So, as we stated in our previous video in Diggle, F x G X is equal to capital f x plus e This we were resuming. So where FX is, the indignant who's in Diggle needs to be calculator. If X is the giggle and C is the constant of integration. The Boston off indignation, mostly referred to as sea, is always there in the result of integration. Now this mainly arises because of the fact that once you take the derivative off a constant , you will always get zero as the result. So when you take the derivative for constant as a function, the own vanishes on and has no impact on the solution, let us make it here with the help of an example. Let's see, we have a function X squared less full, and we take the derivative off this function. So we take the dead Ever do off this function on the derivative since you know by the concept of derivatives that the very bottom of X squared is two x and everybody off for a zero, so answer will be to explosive. So that will be two x itself. And now let us see. Uh, Now, since this is the deliberative, this two X is the derivative off X squared plus four. So by the concert up concept off anti derivative the anti derivative off two X that is the integral off two x will be expert place for no Similarly let's say we have another function X square plus six. Now that is that in every way this as well. So if you differentiate this function also Ah, the answer will again be who exploit Because the different station off X squared is two x and different vision off 60 So the answer will be two x itself. So the derivative of X quit the six s two X and again here the giggle off two x and also we x squared plus six. Since we are just doing the anti derivative from here. Since this is the derivative er just bring the Internet. No, let us say, have you have another function X squared plus and and again we differentiate dysfunction. So if you differentiate dysfunction. Look, this is the derivative. So the derivative again comes as two weeks since the derivative Eric Square is two X and again a very laid off 80 So the answer will, with two weeks on the integral of two x can Also, we reckon, Estes my the concept off Internet directive since he was just doing the anti derivative from here. Now, as you can see from the this, figure that and giggle off two x can be X squared plus four. Also, it can be excludable sex. Also, it can be X squared, plus it. So there are so many solutions for a giggle offer single function. So this leads us to the conclusion that when you take the in Diggle off something, you must add the constant off indignation because there are innumerable e in final number of possible solutions with all sorts off concerts, and all will give the same elevator that is two X anarchists. So in artists, we can ride the in Diggle off two X as X square plus c, since there is no particular reason to prefer one solution out of all these three over the other. So we indicate the entire family off this solution by the Indy Gral by the integral, less this magical constant C So I hope you got clarity about this constant C, which comes in the result off every integral thank you ways. 6. 05 Power rule of Integration: Hey, guys, what's up? Through this video, you will get to know about the most important rule off integration. It is going to the ballot room. This rule will enable you to find the integral off variety of functions like pollen or meals, function involving rules or function with national dumps. So MK practically this rule is expressed like if you have to calculate in Diggle off extra the power and with respect to X, that will be equal toe X to the power and plus one over and plus one plus C on this end, which is the explaining this should not be equal toe minus one It This rule is not defined for Ellen, any equals two minus one, right? So applying this power rule is really very simple. Just you have to add one to the exponent and divide the dumb by the exponent. Less one and ofcourse, the sea will be added in the desert. So let's take a few examples to make it clear to you. So our first example is we have to evaluate in Diggle off X square with respect looks No. Here you can see the exponent is too. So by the powerful. The result will be the signal will be X to the power toe plus one over who bless one. So this will be X cubed or extra power three by three on obviously, A C will be added to the desert. This will be the answer. No, let's take a little bit more complicated, exponent. So let's say we have to calculate in Diggle off Route six. No, you might be doing that route is on explaining. It will do huff. So this can be read it in US X to the power one by two. So the integral off extra the power button by two can also again the evaluated by the powerful. So this will be the result will be extra the power one by two, plus 1/1 by two, plus one. So this will be X to the power three by two. Oh, well, my toe also Or you can write the batter away as to buy three x to the power three by two. Let's see. No, you can. We can take a little more complicated exponents like we have. You have to evaluate the integral off cube root off X Square. Let's see. So again, this can be read rewritten as X to the power to buy three. So we have to find in legal off X to the power bi three now So again you can apply the power room. So the answer will be extra power to buy three plus one over to buy three plus one. So this will be X to the power not to buy three plus minus 5.3. So answer will be extra par five by 3/5 by three and you can write it by to this three by five s to the power five by three. Let's see like so don't be frightened if you find negative exponents also in the function. It is also very simple, as you might be doing X to the power minus and is equal to one by extra department. Okay, Now let's say we have to can play it in Diggle off one by X to the power three picks. How will having really carpet? No since extra bar minus and is equal to one by extra power end. So this one by extra the power and can Britain s s to the power minus and which will be extra the par minus three And we have to find in the middle of extra about ministry so again here also, you can apply the power to the power rule is applicable under negative exponents. Also just the power or the exponent should not be cooled to minus one. So applying the powerful here also, that will be extra the par minus three plus one over minus three plus one. Okay, so that will be extra the par minus two over minus two. Or you can write it, but raise minus one by two. No, since extra power minus two can Britain s one by X square. So this will be minus one by two x squared. Let's see. So this will be the answer and like quiz, you console for in giggle for any function containing an exponent value. So this rule is used heavily throughout in pickles. So if anything is still unclear, I would suggest you to go through the video again or clarify your doubts to look you in a section. Thank you 7. 06 Sum Rule of Integration: Hey, villas, with this video, you're going to learn about this. Some rule off invigoration. This rule tells us how to indicate a function with just some off several times. It basically tells us that we must integrate each time in the some separately and then just add the results together in a battery. Recondite it as the in Diggle off a some off two functions is equal to the sum off there in giggles. Taken separately, the order in which the terms appear in the result is not important. We can state this formally as follows. Let's say there are two functions for fix and G of X, and we have to evaluate in Diggle off ffx plus g affects as a whole. So by the same rule, this will be in Diggle off ffx plus in Diggle off G affects. So it is quite important to realize here that in a function that is some off to or mortems , each dung can be considered as a separate function or even if our constant, um, appears that can also be considered as a separate function. So let's say we have an example here we have to evaluate in Diggle off a polynomial function. Let's say the paranormal function is X cubed plus x square, less x. So we have been valued in Diggle off this as a whole. So we will be applying this sample on. We will be considering all of these three functions as separate functions these three times a separate functions on we will be evaluating the indignant separately. So by the someone, this will be in Diggle off X cube, less in Diggle off X square, less integral off X. So we have just separated out all the individuals. So this is this someone and now you can easily apply the power. So the power This will be X to the power four since three plus one that will be four. Or what? The 1st 1 take me before less X to the power to plus one that is 3/3 on this will be X squared. Let's see, which is the constant off integration So by the Sun room were able to calculate the syndicate. So I hope guys, you have understood the concept well enough. Let's move on to the next rule off integration in the lex. Next lecture seaways 8. 07 Constant Coefficient Rule: Hey, guys, in this lecture, we're going to understand the constant coefficient. Rule off integration. The constant coefficient rule sometimes call us the constant multiplier. Rule essentially tells us that the indefinite in Diggle off see multiplied by FX that affects us some function and see the presents. A constant condition is equal to indefinite in Diggle off FX multiplied by C. We can express this formally as follows See times, FX, the integral we have to help it. This is equal to see multiplied by integral off FX. Like so, the constant coefficient tool essentially allows us to ignore the constant cowfish in in an expression. When we integrate the while, we integrate the rest of the expression. So this means simply we can ate out the constant, um, out off the integral expression while the integrity, for example, let's suppose we have to find the integral extra and giggle off the expression five expert . So the constant coefficient. Let's every help to find find integral off five score. So the constant professional tells us that the giggle off this expression is equal to in Diggle off X square, multiplied by five. So this weekend I test this so In other words, you gun right in the middle of five x Square. We can say you can like this five times in the glove X Square. So another example for this can be weakened because another example is that so we have to find in giggle off to Koszics. So again, by the constant coefficient, all this too is a constant, so we can ate out this, too, from the integral expression on the answer will be two times in Diggle off Koszics. The whole village gathered in legal, of course. Six. Let's think off a function. Who's differential is Koszics or whose different station is Koszics? It's sine X. So the site function sine X will be the in giggle off this or you can see the anti derivative Also off this Koszics. So the answer will be do signings and we will be adding the constant So there isn't so the answer will be who sat next. Let's see. Well, this sums up this video. Hope you have understood the constant coefficient rule. Clearly See you in the next video. Thanks for watching 9. 08 Linearity in Integration: Hey, guys, welcome to the discussion on linearity in integration linearity is one off the mathematics fundamental properties off indication it states that immigration off linear combination of functions is the linear combination off in Diggle. Off those functions, this property encapsulates so simple room, the summer rule and the instant coffee, these tools in a single room. So the some rule and the constant professional we have already covered in the previous videos. Now let's see how the linearity rule is defined mathematically. Let it. Let's say we have two functions f for fix and do tricks. Now consider we have toe evaluate in legal off eight times FX less the times, Jenks, they're A and the r constant these door on symptoms. So how we will apply the linearity So the linearity, rule and absolutes, some room and the constant professional. So first of all, we will split this by the summer to the bite by by some rule, we have this as in Diggle off eight times, FX plus in legal off me times, DX. We can just accept it. These doing deals This one, this one. So by the some rule, we can, um, separate this as it Any specifics in needle? Less indeed. Lovably times, DX hence now what that means is we have further evaluate the needle off thes two fingers, right? No. We will apply the constant coefficient role here with the constant coefficient rule by the constant condition rule, we can take out this constant out of this integration expression. So that will be eight times in legal off ethics, less be times in Diggle, off jigs. So this is the linearity rule off the integration. Let's take an example to see this rule in action. So our first example or slippy are example is we have to eat. Well, wait. An eagle off file X squared less to Koszics. That's it? Yeah. So, by applying, dealing already rule. First of all, we will apply the some of the first part off the linear people. That will be the summer. So buy the summer. We will separate this in legal and this entry. Good. So this will leave in legal off fine X square, less in legal off two mosques. Now the second part of the Lynn at any reputable. That is the constant condition. So this was the someone. No, we will apply the constant coefficient. So what? The constant condition tool. We can giggle these constant out off these constants this fight and two out of the integral expression. So that will be five times in eagle off X squared, plus two times in eagle off Koszics. Now, by the powerful, you can easily develop this integral. That will be X to the power to plus one that is X cubed over to this one. That is three plus in eagle offer. Core sex we have discussed in the previous video. Also, that is sine X. So it will be two signings. And of course, we will be adding me constant city after that after this. So, uh, this is our uncertain. So this was the generic middle. An indication which ankle in car kits and the some rule as well as the constant coefficient I help. You haven't district join me in the upcoming videos. Thank you. 10. 09 Inverse Trigonometric functions part 1: Hello, guys. In this video, we will be talking about inverse stigmatic functions. What no water in West moment of conscience. In worst ignominy, functions are the inverse functions, the deep, regular big moment it functions. These are These are the worst off the regular, pragmatic functions. So the regular diplomatic conscience, you know, like sine X for sex on Ex detector. So now let us try to understand what is actually meant by the inverse diplomatic. First, it's It's Diggle, right angled triangle to explain this. So So this is a right angle triangle. And let's say things is our English, which is under consideration, like his ex on Let's Say, the sides off the Strangler, A, B and C, respectively. It's no let's right, the normal Dignam integrations or the diplomatic functions for this angle in this wrangle. So that will be sign IX R sine X Technimetrics ratio will be a particular upon high partner , so that will be by sea on because it got 63 basic overhype art news. So that is embassy, um, and panics is dependable. One basis, so that this body by it So these are the regular Dignam it. I think these are diplomatic functions for this particular in the X. No, no, wait From these I didn't identities from these ah remembered ratios. If we have to find the angle, that is a measure of this end of X from any of these three ratio. So So how We will compute that. So in order to compute the angle from these identities, let's say from the sine x b Take this This sign that was that site. The right sector. So that will be It will be treated as X is equal to sign in Worse, this is in worse off sine functions. This is not our minus one. This is inverse off sine function X Campbell itchiness signing worst BBC waken say sign affects if sign affixes BBC then that angle is X is equal sine inverse BBC So this is inverse Sinus or malfunction in worse off sine function right? Similarly, from here also we can compute x is equal to because of course, in verse, you might see And from here X is equal to and in an inverse bi, bi bi. And all of these measures are equal in which are equal X so basically thes inverse functions are are used to calculate the measure off the angle here. Right, Right. So So, if you know the results off automatic function or ratio at an angle, then that angle can be computed from the Dignam integration or function by simply taking the in worse off that stigmatic function. Right? Right. So this was this was it for for making making your family or with the inverse trigonometry functions waken Go ahead and see how they're in Borland Needle helpless. See you in the next video. Thanks for watching. 11. 10 Inverse trigonometry Part 2: So we have discussed about the inverse Dignam active functions in the previous video. In this video we will see how they are involved in the in pickles. Let us first start with the first invest pigmented function that this signing for six. So if we do the did he will do off this sign in Verse six. The visit comes as money born undertook one minus X squared. So since the differential off the signing with Texas one upon and the Route one minus X squared, so the integral off distinct and giggle off this thing will be signing for six. Or we can write it more clearly, as in Eagle Off one upon under look one minus X squared, the X is equal to signing or six plus see, which is the constant of integration. Similarly force for course, and we're sex. If we differentiate the course and was X this function, then the derivative comes as minus one over on the road, one minus expert and in the similar French fashion began, See that if differential off carcinoma sex is this an individual off? This thing is course in was X let's see right, see the ad uh, for every integral right. So in Diggle, off minus one over on the roof, one minus x square, the X is equal to causing was X. Let's see Thank. So these were the investigator metric functions sine inverse X and coursing with six. Similarly, we can see for another rig numerical function like this town intersex sort of differential off cannon or sex. What we can say that there really with respect to X off animal sex is 1/1 plus x squared. So similarly, in Diggle off this one upon one plus x square will be this done in was ecstasy, right? So we can write it as integral off 1/1 plus x square. The X is equal to Annan or six plus e. Thank. So these were the three and was most common in Western limited functions. And we use you saw that how they're involved and then we're stigmatic in the indicates right. So similarly, we can just list down in the remaining three investing pneumatic functions and how they are involved in integral calculus. So we can we see that D by D. X off court in verse six that is minus 1/1 plus x quit. So the indignity off minus 1/1 plus x square dx this with me court in worse X. Let's see similarly vegan uh, let's don't the next even though stigmatic function that this second for sex so D by d. X off second words X This is one or what Eggs nicely played by X squared minus one. So integral off one over X multiplied back under load X squared minus one DX that is second worst X plus e and the last in were estimated function that this that will be carsick in verse six. So be body X off caustic in verse six that is minus one over eggs will be played by and the X squared minus one. So the individual off a minus one over X multiplied where X squared minus one and not X squared minus one DX that this Kosik inverse x plus c So nobody this thes investing omitted function and the derivative and in giggles these reserves you confined in the appendix section off our course. So this was it for making your family with all the inverse stigmatic functions being used in the integral calculus? No. I guess you won't be surprised when you come across in West Stigmatic function. See you in the next video. 12. 11 Integration of Common Functions: in this lecture, you will be introduced to the integration off some off the common functions used heavily throughout the calculus. So we're starting with our first reserve. So our first reserve is in Diggle off X to the power. And there this end is any deal number, but it's not equal to minus one. So this is equal to X to the power and plus one over and plus one plus C and the sea is the integral construct. So this is also known as the power with which we have already discussed in the previous video. My second reserve is a in Diggle off sine X. This is minus classics. Let's see off course and the third is in giggle off. Six. This is a call to sign X C now Our fourth visit is in Diggle off annex and this is equal. Do minus log off. More cause X plus e. Our fifth desert is in giggle off cortex. This is it will do. Log off more signings. Let's see. And the proof of these integral of these limited function can be then through integration. Using substitution on this, we will be seeing in a separate video we'll be discussing proof off some of the pragmatic functions. Did you? I have mr here. Like now, the sixth there is a is in Diggle off sex. This is a will do Log off more sex, less panics. Now why? We're taking more Because the but never comes in the log is a positive country. So that's why we wear taking long in here, right? So that is Lakoff More sick, explicit annex. Let's see. And this also can burden through the integration using substitution rule ironed the last one. The last dignity result is integral off Corsica X that is equal to log more Kosik X minus cortex. Let's see. This also can be easily proved using integration off integration using substitution room Our next is there is a in Diggle off A to the power ex dicks that is equal to you. Do the politics over long a Let's see, since we have discussed of in our differential calculus course that difference different station off A to the Power X is a to the about X log. So the integral off into the politics is a to the park ex Bilotti break It is the in worse off different station in that great. It's coming like this. The ninth Reserve similar to this is either the vortex integral, so heated about X. The French station of your politics is it'll politics. So in pickle off into the products will be approximately. So they reserved us into the bar Express E no. Next dessert in Diggle, off one over a square, less x squared that is equal to one by a an inverse exabyte A Let's see. Our next reserved is in Diggle off one, over the road, off a square, minus X squared. That does it Will do. Signing worse. Ex bait, This is Leslie. No are next. Is there? It is in Diggle, off one over a square, minus X squared. That is equal to one. By the way, log off more A pless X over there. A minus, X. Now remember, this law is along with me. Here it is a natural like yogurt. Right? So the desert is coming like this on this can be easily proved using integration by partial fractions. This we will discuss in separate will You similar desert based on integration, using partial fractions is the X over X squared minus eight square also. So this is a cool do when Nowhere the way Log log, Risky X minus, moored off X minus C over. Express a Plessy. There's also and we proved by using partial fractions late. So these are the most used functions when dealing with indeed girls and helpless. So the result off these functions are really important to us. So don't worry about the account of the rules. You'll get comfortable with these once you start to work out on these examples on examples relate to do these acting trees. Right. And I will be always there to help you in case you are stuck some somewhere, right? Thank you. 13. 12 Substitution Rule: hello ways. In this video we will be talking about the substitution rule use in integration. So let's first define Where exactly are substitution rulers So mathematically we can write it as in Diggle off F off Geophysics multiplied by G Prime X is equal to in Diggle f off you. Do you? We're you is it will do g of X now this is a substitution rule on this You might not be getting family or what I have written just now So let us first try to prove this So we know that using Dfx So you is equal to G of X So let us first differentiate What the left side and the exact So this will be d u by d. X is equal to G off deep remix The deeper Mexes differentiation of this function Dfx So this you just to do you is equal toe g brain x dx No, this do you can be replaced by a G Primex DX and you here can be replaced by Jack's. So this will become the strike side will become in Diggle off f off gxe times Do you is G Primex. Deeks So which is equal to the left side. So this we have group the substitution all No. What is exactly a substitution? So if a problem off integration looks complex, something like this, but we're able to see a function who's differential or difference Station is also present in the Indeed here we can say we can assume off that we can assume off a function. Let's say G affects on its in Beagle is present in the function G premix. Great. Then using the substitution rule, we can transform this into eagle this complex and giggle toe a form who's in giggle His family are known to us all. We can easily integrate thereafter. So this this form is ah, little family. This is looking a little, uh, easier in order to integrate rather than this so this can be transformed and do this form. Then there is a function in the integral whose differential is also present in the interior , right? So let us make it here a little bit more clear with an example. So let's take our example. For example, is we have to integrate two weeks. Thanks. Goes off X squared plus one. This is the ex broke so ascherio. Concurrent understanding often giggles This integral is looking out of scope currently. So in order to integrate, look, if we see carefully that, can we take out a function from this and giggle or this problem who's differential or differentiation is also present in this and evil. So thinking about think about it for a while you can see that the function X squared plus one dysfunction is there whose differential that is different station of this function ist two x and that is also present in the problem late. So here we can apply the substitution room. So let's assume Let's see u is equal X squared plus one. So we're assuming this X squared plus one is you, right? Since a different station off this function is present in the integral. So now we have people differentiate this. So do you buy DX? Since we're differentiating with respect to accept this, will we do you ready X on different section of this part will be too explicit. That is two weeks this in place do you is equal to two x geeks. Great, or we can say no normal dumps that dimension off you. Do you is my men as differential off you is two x dicks, right? No, since we know you is equal to X squared plus one which will be placed here and view is equal to two x dx, which will be placed for this product. Right? So no weakened transform this integral from X to the very bill you. So this will be course off you right on this two x times DX that will be Do you So the in eagle has been transformed into a family one. It has been changed from X to you, but the integral is being transformed into a family run. And you already know that in legal off course off you will be sign off you. Let's see. Right. And what was you? You was initially supposed as X squared, plus one. So this is a sign off. X square plus one. Let's see. So you can, as you can see, by using this substitution rule wisely, we can integrate a complex looking in giggle but easily such as the powered off the above discussed substitution. So I hope you guys have industry this rule. Join me in the upcoming videos. Thank you. 14. 13 Substitution Rule - Examples: So in this video we'll be taking some examples on the integration using substitution rule. So our first example is we have to find in Diggle off 1/1 minus two X dx. Great. So we have to find indeed office? No. Uh, normally, you can integrate this, right? So how we will use the integration using substitution, will you? So we'll be applying the substitution rule by first assuming a function whose defense Ishan is president and then giggled. So if we closely look that if we assume this this one minus two x as a function and in the different station off, this will be zero minus two. So that will be minus two and minus two is a constant, so constant can be easily adjusted in the end eagle. And hence we can assume that the different station off this one minus two weeks is present in the indeed. So we can rearrange, erred or adjusted like this. We can write it minus two DX over one minus two x on minus two. Since we're money to playing here. So we're taking out minus two and where it will make it as one great student says This is one. So we're taking I taking out minus one by two. Out from this, integral. All right. No, we will assume. Let's see U is equal to one minus Twix. Okay, so differentiating this thing, we will be getting us. Do you buy? The X is equal to minus two. Right. So this in place do you is equal to minus two dicks. So this this thing is differential off this one minutes to X now you can be substituted here on D minus Studi X Here. It can be substituted. Don't, do you They so the integral no becomes minus one way. Do. No, this is minus two DX is Do you rate? So this is do you over one minutes to X. That goes you. Now you can easily, indeed. Good this thing since you know in Diggle off one by you is Ellen you Ellen Mardi, where Ellen stands for log based e. So this is minus one by to Ellen more you see, So the integral is very easy to calculate when we applied substitution rule. So now you waas when minus two x so we can substitute the value if you here so this will be minus one by two, as in more when minus two X. Let's see. So this was our first example on this substitution. Now let's take one more example. So let's take our second example. No, we're taking a big dramatic example this time. So let's say we have toe find the integral off course off three X plus four. They now how will apply the integration? The substitution rule here. So we have to take off function. We have to assume a function whose differential is present in the indeed all itself. So if you're zoom, let's see you as three express full. Okay, then it's the difference. Yet this it will be three plus zero. That will be three. So it's indeed a three. How will be it will be present here. If we multiplied by one by three and three then we can easily adjust the different she loved this three express for in the integral right. So we are making the adjustments in order to make the integral defense Asian off this function present in the indeed right, so constants can be adjusted rate. So this will be won by three in Diggle off three course three X plus four DX right now. Since do you but be acceptable to three. This implies that do you is equal to three gigs. This can't be good. Traders. Differential off you is three times differential off X right now. You can easily substitute here to change the variable from extra. You know, since we as young this as you, So course off three X plus four will become us a few. And do you was really accept this multi black. But this is equal to you now. You can easily integrate this so this will be won by three. Sign off you since integral off course, you assign you so this will be won by three. Sign off you. Let's see now we can easily substitute you here so this will be won by three. Sign off Three express for plus C. So this was a short example which were over which was covering the trick. Pneumatic functions for being integrated by the by using the substitution. Okay, Now let's take one more example A little trick example on pigna metric function, So we integrated using these substitution. So our example is in Diggle off annex Now, this is just simple annex How you can think off a function here who's differential is also present in the integral. If it if it has, If it has to be integrated using substitution rule, then we have to think of a function was differentiation is also present, right? So how weaken go loaded. So as you know, Annex is nothing but sine X by Cossacks. Right? So we can split it in the form off sine X by signings. They okay? No, you can see a function that this Koszics who's differential that is minus sine X is somewhere president in this integral How is minus an ex president? We can easily multiply outside my minus one and inside my minus one in order to make it one right. So this will be minus sine X Here we take out minus one as a factor on this is Koszics. So no, we can assume that you let's say you is a is equally took Koszics. Now we can again proceed in the similar fashion. We will find the differential a few by differentiating this with respect to X. So that will be do you by D X is equal to minus sine X. So this implies that do is equal to minus sine x mix. Right now we can easily substitute this you and do you in place off this integral to transform this from X to you and make it a really a simple indicator. So let's do that. So this is minus one in Diggle. This is minus one times in Diggle. No minus sine x dx. This thing gets transformed. Toe do right. So this is do you divided by Cossacks and which is you? So again we know in giggle often by u is equal Do Ellen more new right, So this reserve will be minus Ellen moored you. Let's see they so minus Ellen. But was you us Koszics. So this will be minus Ellen. More core sex Class C. So this is our answer. They so these were the basic examples for covering the substitution rule. Right? I hope you have understood all the examples. See you in the next video. Thanks for watching 15. 14 Integration of Rational Functions: in this video, we'll be talking about integration off national functions. So you'll be introduced to the topic off invigoration off for rational functions in this video. Right? So every rational function may be represented in the form off the X over que eggs. So this is a rational function. We're PX and gox Mota. How long is right? So this is a national function. Long legs. I did not back away. Let us say this Russian function can be represented as Why so the numerator part? We can write us in extra the boat in less a n minus one extra the lower end minus one and so on. And the last two atoms can greediness 82 extra color, too. Bless even X. Also in the last three time, these are the last freedoms. Well, where Being extra ballot and plus B n minus one extra. The power in minus one and so on will be toe X square, plus Bieber next class being on not these n and minus one A. Do any in r and B nbn minus one b two b even mean on the Zoeller postures, and these are the power off each other item late Bush I I should I should distinguish these powers that this can be recognized. Let's say this is these older I m in place off and right So where n is the degree off the numerator polynomial and I m is the dele off the You know I'm not the one you normally we're Ennis unknown negative, indigent, right? And also m is a no, no, no negative in digital, right? So if the big d off the numerator is lower than the degree off the denominator, then the rational function is sexually proper. So we conveyed the sense degli off numerator here is And if it is less than the degree off denominator, there does m, then the rational function is set to be a proper, rational function. Otherwise the national function will be improper Russian function. So if there is a rational function, is improper then by dividing the numerator by the denominator, it is possible who represent the national function as us some off the off Polly normal and a proper, rational function. So if the Russian functional lexie p x over q X is improper, then weaken. Divide this numerator by this denominator function and we can find us as MX less. Let's see any under going to be one x work you x, we're amex is the result is a question off Dividing this be expect you x but this is a polynomial and the remaining is the proper production function proper Ashton function they so this is proper action function This is improper rational function and this is ah polynomial so the since the indignation off Paul normals they don't present any difficulties as you can easily integrate using our rule very easily. Right? So the integration of this is not a problem so that the book is the main problem are the main challenge to us That is the integration off this proper national function. Right? So no, it is always possible to write indignant this indignant off the proper rational function as a some off simpler, rational functions And this can ridden by a method called partial function decomposition. So this can be done using this special function decomposition. So after this, the indignant can be carried out. Integration can be carried out easily using the already known matters. So now we will focus. What exactly is this partial function decomposition Blake whereby weaken in degrade the proper national function also. But so let's say we take examples off some off the form off the national functions, some common forms off national function and how we can decompose them in due for most partial function. They So let's take some common form off ration functions. So our first form will be be X. Let's que over x minus a times X minus B. Now remember, this is this is not equal to be here, and this is actually function is a proper rational function were the degree off. This is less than the degree off degree of the new monitors Lesson Agree off, denominator. You can see the view of numeraries one and degree of the non interest Go right, So and also this denominator is being fact raised into X minus C and X minus. Speak so that these two are defectors off the numeric order. Nominative right. So the composition off this inter form of partial function will be off the phone a over X minus a bless me over X minus B, then each other, where the denominator off each other terms is one of the factors So we will write all the factors one by one in separate Come on the numerator, these solar constants which can be simply adjuster by, uh, calibrating from this form. So we will just take out take the else in premiere and we can easily computing numerous monsters. And then when we when you're able to split this form of rational function or you can decompose this former rational function into this form of partial function, it is then easily Indy Global. These two teams are easily indictable. You can easily indicate this and you can insulin directors late. The second form is let's say, if the rational function is off the phone, be express cube over X minus a square. No. One of the factors off this denominator so x minus a is a factor and x minus a square is also a fact like so we will split it in the form off a over X minus a Bless me over X minus a square. Since these tour factors off this you know, Mitt on a or B, you can easily find out by adjusting by taking the else him and then equating this Toby express you right so you can find oh dear A and B constants. And then you can easily integrate from here. They the second, the third form. We can take us B X squared. Bless you X less I Well, we're X minus a X minus B an x minus e no. It is also a proper wrestle function and again began. Decompose it in the form off partial function as a where. X minus. C plus B over X minus v a sealer X minus e since X minus. A expanded speak and expectancy are the factors off the denominated played? No. Our fourth phone is B X square, plus you x plus, uh, bullet X minus a hold square times X minus meat Now again, what? All are factors from this denominator. So when is X minus B others X minus A and X minus A whole square right, So we can split it in the form off a our X minus B less Beehler X minus a. Let's see over X minus A. Well, it's clear. Okay, and then you can integrate this phone simply, And obviously a BNC can be computer by adjusting these left left inside and the right. Answer. They and our final form is something like this. P x square, plus few x less are well, what x minus a James X square less me x plus e. So if the irrational of the rational function the proper rational function is off this form , then what will be the A composed partial fraction? No, no, that this x squared plus b x plus e. It is, in fact, right, in fact, risible right. So it cannot be decomposing toe factors, so we can factory. Is this right? So what? All will be the factors off this denominator. So when will be X minus eight On the other will be this whole X squared, plus the express. See right? No. When we decompose, um, affect er which is in itself off of power more than one. Then we will be writing in the numerator off the phone. If it is like it is all the foam, the X squared plus B x plus e. Then we'll be writing in the, uh, numerator off the form X plus B. There we will be making one power less from the this factor. If it had been excluded. X Q plus B x squared plus C x squared plus D. Wade. Right, We will be writing like X squared plus B experts see over this factor. So remember this thing and of course, the next will be see over X minus. So if the fact that is in itself like here it was expense a whole square. But the factory itself, without the power was off, the was off a single single, but right, so then we were simply writing us. See your X minus the square. But if it is in fact risible on it is off the form X squared plus B express. See, then we will be writing as X plus B over that factor. Okay, so this will be X plus B over X squared plus B express C plus. This is normal. C C over X minus E. No, you can go further to integrate this in this part instead of this part late. So you will get more clarity, uh, with the upcoming video than when we will be discussing some examples based on integration using partial fractions. So this was it for the heretical part off be composing our rational, proper national functions in the form off parts infection. Thank you for watching 16. 15 Integration of Rational Functions - Example: So in this video we will see examples off integration off irrational functions. Now, as we discussed in the in the previous video, integration off rational functions is mostly done by partial fraction decomposition. Then the degree off numerator is less than the degree off denominator. So we will be considering those cases here in examples. So let's start with our first example. Our first example is integral loss three X plus 11 over X squared minus X minus six late. So, as you can see, the degree off numerator is less than degree off the nominal degree of numeraries one on degree often denominator is to a degree of numerator is less than degree off denominator. So here we can apply the partial fraction decomposition and how we will able to go apply the partial fraction the completion. First, we need to factor as the denominator for that. So let this factor is this thing. So if we have two factories, this this will be X squared minus X minus six. So, by the middle time, by breaking the middle term, we can you make this as this product is minus six, so will break us minus three and plus two So that will be the product will be minus six, so we will be breaking the madam as X square. This will be minus three X plus two x minus six. So when we do the factors, this will be X minus three less twice x ministry. So the factors will be X minus three times X plus two. So these are X minus three and X minus to express to are the defectors so relating the above Integral we see this is three explosive 11 over X minus three times X plus two. So now, by the partial fraction decomposition we conveyed, we can write three x plus 11 over X minus three times X plus two as a constant, a over first factor. That's another constant. Be over these second factors. Now, if you're able to find the constants A and B are integration task will be quite similar. Let's go on to find these constants. Now we will find the concerts so we will be able to find the constants when we equate. When we make the denominator is equal than the obviously The numerator is will also be equal, right, So making this denominator taking the common denominator. That will be X minus three times X plus two. So this will be eight times X plus two. Let's be times X minus three. Since the denominators are equal, we can equate the enumerators as well, so this will be three x plus. 11 is equal toe a X plus two way less BX minus three B. Now let's make the provision off X it great. The conditions off X and the o vision the constant provision equal. So the coefficient of X will be a blessed be on the remaining will be way ministry be so by equating the conditions off the domes we see a plus B will be equal to three. So this is exoduses our first equation and way minus three B. This is a call to this 11 late Mrs story. So this is our second question, right? So now let's going forward to I saw these two equations, so but I was our first question off equation waas a plus B is equal to three and second aggression was the way minus three B is equal to deliver, right? So we need to eliminate when very evil were preferring the other. So how we will do this, we will multiply this bite too. So this will be six so way. Plus, we can write it in a better way here. So this will be a place to be is equal to six not just subtract these two equations So this gets to it gets cancelled with this and this will become five b is equal to five, right or sorry, this is We're subtracting this so this will be minus five. B is equal to fight to descend Plays busy will do my next one Right now we since my first location we knew that a plus b is equal to three right, So substituting the value of be here That means a minus one is it will do three we get is it will do for So now we are able to derive are constants A and B whose values are four and minus one respectively. So let's just substitute this to find the integral So are in eagle waas three x plus 11 initially, then giggle waas three express 11 over X squared minus X minus six This big game as this was a over next ministry Let's be over X plus two. Right. So it was equal to four and B is equal to minus one. So this is equal to in Eagle off a over experienced E that this Ford over X minus three plus in Diggle off be that is minus one over X plus toe. They or I didn't know better weapon battery. This is four. So it will be moved out off the integration. This is the X over X minus three and the rest minus indignity X over expressed. Oh, since you know in Diggle off X one over X plus B is Ellen X plus B. So this will be four times Ellen More x Ministry minus Ellen More X plus two. Let's see. So this is our answer for the first example. Now let's take our second example. Our second example is integral X squared plus four over three x Q. Bless four X Square, minus four dicks. So since the degree off new monitor that is to is less than the degree off denominator, which is three. So here we can apply the partial fraction become position easily. So first of all, we need to fact lies the denominator for that. So as you can easily see, X is a common factor from all these freedoms. So we can write. Write three x cube, less for X Square minus four x as x multiplied by three X square plus four x minus four. Not this can be further fact raced by breaking this middle term since the product off this and this is my nest. Well, so we have to break this in the form off us in the form of home says that that work is minus two yl X square, so that will be six x minus two x So let's break the momentum. So this will be six x minus two X minus four. So now it began factories. This this will be three x x plus three. What a story. This is X plus to minus two expressed. So our final factors will be X multiplied by three X minus two multiplied by X Man X plus two. So this partial this think and readiness eBay First factor X Let's be by second factor X plus two. Let's see by third factor that this three X minus two right? No, let us make the numerous denominators equal, you know, equate the Newman it is, that's what, mate. So our numerator is on the left side that does X squared plus four. And on the right side, the numerator are a X plus two multiplied by three x minus, two less Be my supplied by exit to three x minus two. Let's see X multiplied by its place toe. No, again, we will be, ah, making the coefficients off each other times equal. And then we will be able to find out by making three questions the values off A, B and C so for this example escaped escaping that part for you guys to solve yourself and I'm going to the final solution. We'll find the values off a B and C as a is equal to minus one physical toe when by do and C is equal to five by toe. So have I have faith on you guys that you can easily solve this these three questions, as we did in the previous example, so finally were able to find big the integration, this one in the form of a partial fraction decomposition as a by integral off a by ex that this is minus one. So minus one b x by X Bless B is half, so that will be no be was. Be over. Express to right so b is half. So that is one by two. The X over X plus two. Let's see, she's five by two and giggle 1/3 x minus two. So, finally, our answers Tom the as minus Ellen More decks plus integral off when we express toe is Ellen More express to and similarly in Deal off one by three x minus two Will be Ellen more three X minus two Divided by the coefficient off X since here, since if we were differentiating, then would then the different station off. This will be three. So now we will be dividing by the writing by three. So that will be five by toe, multiplied by one by three. Ellen moored three x minus two Let's see. So the final answer comes as minus Ellen. More decks less when my dough and in more express to Let's fight by six Ellen moored three X minus two a. C. So this is our solution. So, guys, these were some examples on indignation off rational functions by partial decomposition method where the degree off numerator is less than the degree off the normal. Then you can easily apply this matter. So this was it for this particular video. You join me next time. 17. 16 Integration of Irrational functions: Hi, guys. In this video, we will study indignation off irrational functions. First of all, let us understand what our ration functions functions which are not rational. Of course, that means they cannot be written in the form off we make. You were Be back. U boat, huh? All normals. So such functions are oil s international functions. This means the function involves a radical, which means a term off the form off x. Or let's say any other function to the power MBN, where this m by n is rational and and here is not equal to one. So in that this is if the function consists something like this, then that function is basically irrational function. Let us say something like this a function off something like this. Okay, so here this excess to the bar one by toe, Xs to power one by threes existing. It is off this phone. So the radicals are there in the function. So that's where this function overall is an irrational function. No, no, indeed. Great and rational function containing a or X to the power in my in or if x to the power M Brian, we make substitution. Let's say anarchist, It's containing extra power when by do an extra borrowing by three So we need the substitution You is a Quito X to the power one by n the and is the least common Will people or the Elsie um off all the denominators off the of all the fractional powers which are appearing enough function now what are the attraction powers which are appearing in dysfunction? It is extra when by do here and one bite three here. So the denominators off the fractional powers are do on three. Now what will be the Elsom off these denominators? So l see him off two and three. That will be called to six, which means we have no assume you is a girl do X to the power one by six. So if we make such a substitution and dysfunction, we will be able to integrate this irrational function quite easily. The documents in this function we have toe a musical do X to the power when about six So putting the above substitution in this irrational function, the irrational function can be transformed into a rational function. So once this is actually function, it's transformed to this the any arrest, any rational function, then it can be integrated for the by using the partial function become Bazian as we have discussed in the integration of rational functions. No, let's take some examples in the next video who make this concept more clear. State Yoon and focused. 18. 17 Integration of Irrational functions - Examples: So in this video we will be discussing some examples on integration off irrational functions. So our first example is we have to evaluate in eagle droop over Express nine over x. So since you are noticing that this system is off a radical sign, that means this overall function is an international function. So we have to make a substitution in order to make this international function in a rational function. So since there is only one which is under the radical sign, then we will just assume that is equal. Duke you. So that means you is It will do in the root, explicit name. Which means explicit name is It will do you square right? Or for making X decide, we will make this as acceptable. Do you squared minus nine, not differentiating both sides. The differential affects d X is equal to do you differentiation of you square is to you multiplied weather differential a few that this do u minus two different station off nine is zero. So dx isn't going toe toe. You'll be you great and usable to this thing. So let's substitute the X and the love X in this function so That means this isn't goingto now, since this is you so integral you Andy exists. Do you do you over X access, You square minus nine. So this is equal to two names. Integral. You square over you square minus name late. So that means this is ah, wrestle function. As you can see, they So now we have to go in for the to integrate this But I still function. No. Now, since this is you square over, you score minus nine so we can write this as you square minus nine last night since we can split Wyndham according to the numerator, So that will be twice integral. This will be one and plus nine over you score minus night late. So we have done this in order to make this this one in the form off a proper lesson function. No, we can apply the the rules which you have discussed Ah, as an identity for invigorating starts going off expressions. So this is easy in Diggle awfulness. So this is in big alof one. Do you see that will be you bless nine times in Diggle off do you? Over. You score minus nine. This is not right. So this will be Do you bless? 18 times in Diggle Off one over. You score Afghanistan now This is off the form X squared minus a square. So the integral for one over X squared minus a square. Waas. So this is you. Square ministries were right. So in general is one, By the way, so is 3 to 1 bite. Women's six Ellen moored X minus a over explosives with in head. In our case, it will be U minus three over New plus three late. So this is the integration of this particular thing. When might be a little more expensive. What? Ecstasy. Right. So this is one by six Ellen more you ministry with you. Press tree. No, this is Let's see. So this finally is in going toe. Do you know what was you? You was substituted as on the road Express name Bless. Three hour 18 by six. From here we have 18 by 60 Physical to three. So three times Ellen, More humanistic. So us on Route Express nine. So we're just substituting back the original variable in order to compute defiantly incomes off X. But this is you press three So this is under root. X plus nine plus three. This is all over warned under morning. Let's see. So this is our final solution to the first example. So now let's take another example. So our second example is we have to integrate X to the power one by three last one over X to the power one by do minus one. No, here are totems basically extra the power one by three and x to the power when by two which are under the radical sign because their powers at national here it is won by three and here it is one by two. So, as we discussed in our when we were discussing the approach for integration off irrational functions that if there are more people functions which are having rational powers, then we have to make a substitution off. Let's say very well you is equal to X to the power one by and hear the function is X right . So we have to make that function is X x here. So we have to make substitution musical two x to the power one by n where n is the Elsie um off the denominators off all the fractional powers. So what are the denominators off the fractional powers that is three. And so the LCN will be six. That means and will be equal took six, Which means you physical toe It's the power won by six. So this is our substitution, which we will be making in this particular question. In order to make this investment function transformed. Blue dress and function Let's do this substitution. So let's say we have you is equal to X to the power on May 6. It means X is equal to you did about six. Right now let's different shit on the both sides. This means the X is equal to the differential off you do About six is six. You did of our six minus one that this fight times the differential off you. So this means the excessive will do six year to the bar five do and excessive will be. You did about six. So that's subscribe to the values off Ex nd X In the about question to me, this transformed in the variable you so since X is equal to you did a power six so so are question will be extra power one by three. Last one over X to the power when by two minus one will become So this was an eagle was under DX. So this will become this is U to the power six by three. That means you square plus one. And this is u U to the power six. Bite this. This means you minus one. Right. And, uh, the X was equal to six u to the power five You. So this is multiplied by six. You did about five, do you? Right. So this is the final integral. It's you have to develop. This means six comes out of the indication, since it is a constant, which means you did about five. That means if he multiplied this with this old thing, that means you've stood about seven. Plus, you did about five over you cube minus one late. So now you consider you noticed that this is up is not a proper fashion function. We have transformed our rational function to wrestle function. But this is not a proper, rational function. Since the degree off new military that is seven is very much greater than the degree off the normal privileges. Three which leads us to a conclusion that we have to divide this national function. That means we have to divide this. You do about seven place with about five with you, Q minus one. So whatever will be the question that will be a dumb plus. Whatever will be remaining off the form affects over GX. So this will be our proper rest in function, which means after the waiting, the remaining But I still function will be, says stack the degree off the numerator will be less than the degree off denominator. So integral of the first part will be very easy. Sensitively, it will be up normal polynomial. An indication of the second part will be done by the partial fraction become position. Since this is since this is a proper, rational function, this will become a proper, rational function. It will be a proper the national function. That means you can easily apply the partial fraction become version technique which we have discussed in the invigoration off rational functions. And because you can integrate this particle a problem. So I hope, guys, you can try your hands on this question yourself. So this was it for the examples off indignation off irrational functions 19. 18 Integration using trigonometric identities: Hi there. In this particular lecture, people see how we can make use off the common pragmatic identities to make the integration off hard to integrate problematic immigrants quite easier. Right? So let this first discuss what are the common pragmatic identities that we will be using? So first is signed. Wicks is equal to you. Say Nix classics. So generally, when we can see in legal immigrant of this type some sign x koszics so we can use this thing Sign two x This part on you know that integration off sign MX or sign an X is quite easier has compacted this so you can meet make use of this identity to make that indeed and simple. I was thinking this cost two x is equal to course quicks minus signs critics. So this is the identity. And if there's no that Oscar explosives work physical the one so you can write this thing in terms off science correx that will be one minus two science critics and similarly, in the form off course, Kerekes if you is this with one minus cost works. So that will be two course critics. My nessman. So if you find the indignant in the immigrant some science school, X tome or Costco system. You can use this identity to substitute this cost two x in place off those things and make the indigenous indication quite easier. Great. No third entity is signed. Three X is equal. Three say nix minus for Saint Coolpix. Now, when you will see some, um, getting signed. You, Becks, in the Indian, you can use this. I didn't mean to make the integration simpler. Next is do sign E. Cosby. This is a cool do sign a plus B less sign a minus, bi. So you will use this opportunity when you find the indignant, somewhat off this type signing Cosby. So you can use this identity to split it in the form of these two terms, and then you can easily integrate thereafter. The fifth identity similar to this is Go say, sign me like so that will be sign a presby minus sign a minus bi. Right. So no, this six. That entity similar Toby about the previous two is to Kasey Cosby, and that is equal to close off a plus B less cause off a minus bi. So when you see immigrant somewhat of this type courses. Cosby. You can use this identity to make the integration single. Okay, the next is next off. This to the final of this type is due signing signed B, and this is equal to cross off a plus B minus yourself a minus bi right? So these are the few pragmatic identities, but you can make use to make your immigrant look very simply simpler. Right on. You can and even thereafter, were easily. Now, let's see in the next video some examples to see how we can make use off the above identities to make our task off Finding the integral quite easy, stay tuned in. 20. 19 Integration using Trigonometric identities - Example: So in this video we will be seeing some examples for integration using pragmatic identities . So our first example is we have to integrate course critics bx right. So we will see how we will use make use off that ignominy get entity. So let's recall back to the identities which we mentioned so we can see from this identity cost. Two X is equal to look or school X minus one So we will make use off this identity. So anyway, because those two X is equal to do cost critics minus one so you can weaken, rear ended really injured and we can see Costco Rex will be is it will do, of course, two x plus 1/2. So no integral of parts Quicks is impossible. Normally using the power of a zoo know that you can't apply the power to layer because it is applied only for as your break expressions algebraic functions. So we will substitute this value in place off this to make our indignant simple So are indignant. Will be I will be imperial course two x plus one or were too knicks. Now this constant one by toe will come out by the constant factor rule Onda the integral will be cost two X plus one. So now we can easily integrate No, In big ALOF prospects is say Nix. So in legal off course two x will be signed. Wicks, buy gold. So it is just like opposite off, changeable and changeable. We were medically playing with the official with the inner function here, dinner functions to X so river multiplying with the differentiation off this here in in Beagle we will divide by that constant in order to make the And today we're off this equal to in order to make this derivative equal to this since anti derivative of this is signed to Expecto. So you can easily specify that you when you differentiate this this will be to cost two x or toe So that recourse tricks. So we will remember that this, like when we apply the change will in different station. We want to play with proficient that is to here. So it's similarly an indication I'm really worried about that, right, So this will be the signal will be signed two x my toe and then little off one is X they so this will be signed. Blix by four plus expert too. Plus see, it is the integral monster. So this is a result. No stick another example. So our second example is in Diggle off signed Twix cost three x late. So we will see how which identity we will be using here. So the question was in Gigolo signed two x cost fix. So that is a form signing Cosby. So we will apply this term. Sorry. This identity. So that will be do signing course we Is it going to do so? We're right here. Houssaini Cosby. Is he cool too? Sign off. A plus B Less sign off a minus bi. Right. So we will see what our values off nd So a here is two x and me is three x So we will substitute these values here and find the desert. So that will be so. Do sign two x three x. So who signed to Xcor streaks? That will be equal to sign off a plus B A plus B is five x plus. Sign off a minus bi. That is two X minus streaks that this sign off minus six. Now you guys must mean wing. That sign off minus X is equal to minus sine X. So this thing will become Sign off Ibex minus Cenex. Now we will substitute this in place off this and then giggle. So this was to sign to Xcor streaks. So signed two X clothes three x will be won by two sign five x minus signings. So, in order to invigorate this thing, we will use this thing now to integrate right? So are in legal becomes when they do and giggle off one by do sane five x minus. Say nix. So this is what we have to integrate. So one way to is a constant, so it will come out of the integration by the constant factor rule. So the final integration is signed five x minus Cenex No integration off sine X is minus for six. And as I explained in the previous example, we will divide by the off a vision in order to make the indignant equal to 90 days until it will be minus costs five x by five. So, when we differentiate minus cost five x by file. We will get this letter so that will be in legal will be so this will come by practice that you have to be worried by the profession. Or you can just imagine this as the inner function. And when you're Differentiate was in general, you will multiply with the different station off. Five extract will be filed. So here you will be dividing by the differential coefficient. That will be five. So minus sign five x my five. And in divisional Sine X is minus course annex. And this is science minus already here. So that will be positive. Course I necks. And of course, a C in little constant is obviously a great so final answer will be minus one by 10. Who signed five X plus one by do course I next. Let's see. Great. So this is our answer for the second example nihilistic for example. Sorry. For example is we have configured cost two x minus cause Tau Alpha, we're con sex minus or sell for the selfies. Also another variable. So we have to integrate this thing. No, which identity we will be using here. So, as you can see in the numerator, you have forced to extend course that cost over. So if you want o make use off some identity here. Normally you can You can do some do anything in order dated. So we have to use the tournament to get entities. So we know that cost two X is equal. Do who goes could X minus one. So here's what we were doing. We were using this thing in place off this to invigorate. Now here we will make use off this fight in order book, make and limit this thing. So let's substitute. Let's first Ray to see what happens when we substitute Cost two x as to course correct minus one and cost. Two wildfires broke Oscar l four minutes one. So let's substitute that. So I didn't feel it will be course critics minus one minus Ouko Square Alpha minus one. And this is who would So this is aware course six minus course. Alfa late. Not this one. This is minus one and this is minus off minus one. So that will be cancer. So this will be cancer and this will be so this will get cancer right? And this will be to Costco X minus. Costs could open. So too will come out as common and that will come out of the integration as well, since it is a constant. So that will be toe or school X minus Qasr El far over course, X minus casal for no, this is something off type A square minus B square. So since you know that a square minus B square is equal to a minus bi multiplied by a plus B they so we will make use off this identity. So in eagle will be Do you multiplied by now? That was course Kerekes minus cost squirrel fur. So that will be a plus B course. Express yourself for multiplied by a minus. Bi that discourse X minus causal for on the numerator waas course X minus or sofa. Okay. And this is Deeks. So we can cancel out these totems called six minutes yourself for provided core sex. Should not be acquittal, of course, in for So this. We're assuming that cost X is not equal to call so far, or you can assume that it is going in the question itself. So if it had been caustic concepts, if it would have been equal to course, Alfonso, that's will be zero by zero, said we That would be how that would have been in indeterminate form. And we can't cancel in that case. So assuming for success nautical The causal phobia canceling, cancelling out these totems from numerator and denominator So final and eagle becomes as in Eagle off. Koszics That score so far complied back. Sorry, this is Koszics. Bless. Course l for Okay, So we know now the very we're integrating with respect to X So in big alof Cossacks will be sine x Now this causal far. This is a constant air with respect to expect since wearing integrating. So this will be treated as a constant right, so constants So this will be integral to times integral off Koszics dx less twice integral off Final four the X So this will be simply to signings later? No, Since this course over, this is a constant. With respect to X, it is a constant right. So this constant will come out of the integration. So that will be to yourself are And this will be one. So the final answer will be new signings place an integration of oneness X since different station off excess one. So in eagle of one will be X. So this would be to x yourself for bless, See, which is the integrate one street. So this was a little trickier that you have to understand that this carcelle for system If we're integrating with respect to X, the score sulfur will be treated as a constant. I didn't mean far here deal for then it would have been taken as a variable. So this was it when we were Ah, this was it for the third example. So these were some examples on the pig pneumatic identities. When you have to make use of that ignorant content is for integral. Join me in the next video. Thank you. 21. 20 Integration of Special functions - Part 1: welcome for this video. You really get to learn about immigration off some off the special functions. We will also see how these results could be derived from already known matters as we have discussed so far. So let us start with her first result. So our first special function is Vanover excluded minus hey square. So if we have evaluated bigger love one over X squared minus eight square. So the result is one. By the way, Ellen on Ellen isn't log e mourned X minus e over X plus a plus C. And the second dessert or the second signal of special type is in eagle off, one over a square, minus X squared. And the result is one, by the way, Ellen a Plessix Well, what a minus six. Let's see now let's try toe prove one of the above reserves and you can obviously proved the other at the result similarly. So let's tryto prove the this reserve. So we have integral the X over X squared minus a square. No, As you guys know the identity a square minus b squared. So we will be a playing so the this identity. So this will become X plus a multiplied by X minus e Know how you can split these totems. So if we split split these totems than easily you can integrate one over explicit and one over X minus. So how we will split these two in the denominator. So whenever there are certain certain comes in the denominator when the defense off the totems is a constant, then you can separate these totems. So this is a trick which you can use here. So let's see how district amused. So as you can see the difference off these totems explicit an X minus eight is well, since you subtract x men, etc. From this we will get the result as toe so but we will do We will multiply and divide by this difference So that will become something like this late. So this is when I worked with in Eagle Way Over explicit When you play by expectancy, right? No, If you step if you write the difference in that form only that explicit minutes expensive it was to it. So you can write this Do it as explicit a minus x minus c late. So, by writing in this form no we will be able to separate these totems. So if you see this dumb here, this explicit gets country with explicit on the, um, becomes I know what X minus e. Right. And if you consider this one, then this X minus A gets canceled so that, ERM will be won over. Explicit. Right? So it was it become It has become no easy to integrate this thing. So no in giggle off X minus a or only a little off type X when over X plus See their season One string is Ellen X plus C Bless the Let's this even which is the integration of Constant . So in legal of these types is this Ellen Extra c went over. Experts see integral off when our expertise and an expert see So we will use this here to find an answer. So our answer will be Ellen more X minus C. Since the constant was minus E minus Ellen more explicit. I know this leave us see will be at it So no, you guys must know this wizard. That log M minus Logan is log emmorton. So Ellen X minus more next minus e over minus Ln x plus more explicit will be Ellen more X minus a divided by explicit because log Ellen M. Minus Ellen n is equal Do Ellen M by N So this is a lot property which we're using here. So finally, we have were able to come to the result off the first first reserve. So that is equal to one over this indeed, often over or excrement say square is equal to one over away. Ln X minus a over explicit they So similarly, you can prove the second result this one by using this similar trick which are used in the this proof and you can come to this desert so we will see a few more in giggles off special tape like this in the next video. 22. 21 Integration of Special functions - Part 2: So in this room you will see some more in Biggles. Off special types, sort. Hi. Special type is in giggle. NeeIix over there on the route X squared, minus a square. This is a Quito Ellen moored X plus under X squared, minus a square. Let's see. And this Ellen is logged. This and our fourth desert, which is similar to this one only that is an eagle off under root X square less a squid. So the result comes as Ellen, moored off X plus under rude X square, plus a square fluffy. So let's allow not tried to prove one of the results. So let's prove this resort. And similarly, if we can prove the fourth dessert, you can use the similar approach to prove the third reserve. So let's prove this reserve the fourth dessert. So that is an eagle dx our a little X squared. Bless a square. No. We have to eliminate this and the route so as to integrate this integral so we can use make use off a big dramatic an entity here. Something like this one. As we know one place stand square. Thida. This is a cool do six square teeter they so if he substitute here at this place X equal to eight and teeter, then this will become a square, plus a square transport feta. So if a school gets common, then the bracket it becomes something like this and weaken substitute one place stands quota as 6 40 Right? So let's do that. So if we substitute ex acquittal eight Antigua, then what will we be? X? So we have two different shades with respect to data here. So we will be differentiating this with respect to theta. So that will be a six square Peter. So the expert Geeta is equal to a sex party later. This implies E X can be written as a six square teeter Nikita. Okay, so we know X hair and the excel. So we will use these two desserts and substitute here transform this individual in the form of teeter. So let's do that. So integrally excess a six square Tita, Did he go over on the road off a square? Let's a square dance. Quit data. But that will be a square sex court data. And this is course expert Peter is in direct route, so this will become a and this will become SEC data. And this a gets cancer. Really? To see on this sector gets cancelled with the sex court feeder as and it will find we as integral off sex it only big no, in legal of sect leader. So how we will find indeed a lovesick freedom. So in signal of spectator or you can say integral off sex only We have mentioned in our one off our previous videos where we discussed indeed ALOF common functions We mentioned result but stated that integral off Sect X is equal to Ln more sec expressed annex So here the integral will be Ellen more sick Peter bless 10 people fake So use the result for the integral of spectator and plus C obviously will be here. Now we know that X equal to eight and Peter So we have to substitute this spectator intended them back in the form off X. Just give the final result North 20 ties. It will do X by a late from here. So how we will find spectator So as we know one plus 10 square Peter is equal to six quart Peter So Sex spectator will be cool do under Route one plus and squared, Peter. So this will be Underwood one plus x squared away in squared. And this will be and they don't x squared, plus a square if you take it. A square is a common denominator. So this will be under your desk or is a suspected. That will be this and And he does this so we will substitute values off. 2010 Spectator in the individual. Thank. So this will be Ellen Spectator was under root. X square plus a square, My A and empty the waas. Explain, Plessy. So let's take a common denominator here. That will be Ellen. More X plus under root X square less a square, Order A Let's see No by the property off log M by n is equal to Ellen M minus Elhanan so we can use the property here. So this will be Ellen more X plus Underwood X square plus a square minus l in a plus e. Now you can notice that is a constant, so l any will also be a constant. So this constant Plus this constant can be written as a new constant C. So final result will be Ellen More X plus under root X squared plus a square. Yes, See? And as you can verify from our dessert that this only was the desert for this. So we have proved this is er so similarly, you can use the concern the approach discussed in the proof to prove this is a lesbian. So whenever you come across in eagles off such types, you can use these results get through the solution quickly. 23. 22 Integration by parts: Hey guys, In this video, we will see how to integrate When the immigrant is a product off to unlike functions or this condition when they integrate itself is a product off two different functions. We use the rule off integration by parts. Well, first, see the proof for this and literally with some problems, begin with understanding. So let's start with the proof. Let's say there two functions you affix and view. Fix these both hard, defensible functions on what are dependent on a single variable X, then buy the product off. Direct rule of different station We know that debate. The X off UV is equal to first function multiplied by different station off second function bless second function multiplied by different station of fast function. So this is the directory for dispensation late. Now let's reinjured. Uh oh, are we can use the integration here. Let's integrate both the sites so integrating would saves. We have This will become UV right. Since we're integrating the different station of this so that will be you itself on. This will be in legal off you multiplied by D V by D X plus in legal off the Deuba mix let Israel injured, do you? Let's take this from here to the left side. So that will be in legal You d v by d. X or we are integrating. So we just need toe. But in general the X so this will be you d V by d x Integral is equal to you ve minus So this time I was there. This will be minus we D u by D. X indeed, Mrs Deeks. So for Let we have to calculate in, giggle off productive two functions late. So if you assume this as our first function on this as our second function, then we can we will be able to drive a formula for the integration by parts. So let us suppose you as the first function on d v by bx as be setting function. Right. So let's say our first function is ffx and second function is geo fix, right? So what were we were able to derail last step that was in Diggle off you D V by D. X is equal to u V minus the integral off B Do you buy dicks? So it feel like this as the first function this is the second function FX and GX respectively. So let's try to find the remaining bombs. So if d v by D. X is the second function, what will be be So you is FX d V by D. X is gxe We have simplest, right? So what are the remaining terms we need to use calculated? We will be What will be We will be integral off Jack's And here we is calculated so d u by d. X is not calculated. So do you, Buddy X, from here will be different station of this. So that will be f premix. Brian means the different vision of dysfunction. So if this will be a premise so now let's substitute the values from U and V two effects index so this will become the first part will become so this will become as in giggle FX multiplied by GXE in the middle of this is equal to ethics And that was this was be was indeed a lawful GXE. So this is integral off NGX minus integral off me times the U N E X So V is we have mentioned in the middle of DX and you weigh the excess of premix. So this will be if Premix Times in Eagle Off GXE and this whole is being integrated, right? So how we can write it as this is first function and this is the second function. So we can write indication by part as first function into integration off second function minus different station off first function integration off SEC multiplied by integration of second function, and this whole is being indicated. So this is a very short Henry off writing this integration by parts as you can easily remember this this thing means in legal off that function. And this thing means different station off a function so weaken right in. Or we can say that integration by parts is an indication of the product. Off two functions is a choir that is equal to first function multiplied by an eagle off the second function, minus in, giggle off differentiation off the first function multiplied by an eagle off the second function. So please guys do remember this, that this whole thing is under integration, right? So this was the integration by parts, the rule for integration by parts. Now let's take a few problems to see this concept in action. Thank you 24. 23 Integration by parts - Example: So in this video we will see some examples on integration by parts. Let's first reported what was integration by parts. So that was if you have to calculate in legal off product off two functions that will be equal to first function multiplied by in legal off second function minus integral off different station off. First function multiplied by integration off second function This hole is under the integration like this indication. So this was our indication by parts truly No, these are This is the first function and this is the second portion. Know how we will assume which will be our first function and which will be our second function. So there is a particular criteria by which we can choose our first function and automatically the second will be the second function So cried area is named as I lied That s I l e So this means inverse functions. L means logarithmic functions a means algebraic functions. He means signal metric functions and e means exponential functions. Right? So what is the criteria? Great idea is that we go in this order starting from this This is this is having the first priority and this is having the last priority. So we go in this order, do little mine. What will be our first function? Right. So the if we have inverse as the in the product So we select and with the first priority Then after that, this has this is having the priority. So this is the order of priority off selecting the first function. Right? So now let's take an example to see the integration by parts in action. So let's say our first example is in Teagle X Classics. So now how we will select the first function. So as we mentioned, Islet is the great area. So what are the functions we have? This is the algebraic function. This is a or this is the ornamental function. So what? What what will be dissected as the first function since we know it comes first and I'll eat . That means this algebraic function will be selected as the first function and automatically the other will be selected as the second function. So this is the criteria. Oh, this is the first function. This is the first function and this is the second function, right? So now let's apply the integral by parts. So in Eagle off X, koszics will be first function that is X multiplied by a giggle off. Second function that this this minus integral off different station office function Know what is different section of X that this one multiplied by in legal off second function and this whole is being integrated. I think this is the whole motivation. So what will be integral? Koszics That will be signings. So this will be X sine x minus in eagle. Of course, it's here again will be sine X so this will be integral off sine x. Right. So this will be coming as X sine X Now, what is integral of sine X that is minus co sign X and this is also miners are already here , so minus and minus co sign Next will be positive course annex. So the answer will be X signings Prescott sex. Let's see. So this is the result which we are able to die where the integration by parts Let's take one more example. So our second example will be in Tickle X do the biotics. So this is our 62nd example? No. Again, we will apply the integral by parts here. So Bill is like, what will be our first function? Right. So by the concept off islet, you see that this is the algebraic function. This is the exponential function. And by our islet we see that a comes before e. That means we select as a brick function that is X as our first function and e that exponential function that is eat about X as our second function. So now let's apply integration by parts So this will be in legal extra robotics bx This will be first function that is X multiplied by in eagle off second function that is this minus integral off different station off first function Nobody x off X multiplied by in legal off people Power X on this hole is being integrated, right? So let's going further. We will your that is it. So this will be X into the bar x minus Anybody except one X will be one. And in the middle of it about excess, it'll bar X so this will be one times it'll about extract will be e to the buttocks. So finally it will be a to the Power X gigs and we know that in eagle a few did about excess. It'll bod x x itself. So this will be X e to the war X minus. Either the world X Class C or we can take it over X common. And this will be X minus one. It about excellently prepared. Waxman is one the sea. So this is our reserve. So, guys, this these were two examples on integration by parts. And this rule is a very useful rule when you are stuck how to integrate some functions. But this rule comes and really comes as a handy tool when you have think about integrating unlike functions or product, often like functions late. So this was it for this video. Catch me next time. Thank you. 25. 24 Miscellaneous Examples - Part 1: In this video, we will practice some Muslim is examples by which we will try to cover the important topics of integration discussed throughout the course. So let's start with our first example. So our first example is we have to integrate. Do over on the route nine minus four X quip late. So this is the question now how we will proceed to solve this question. Nice again. See, this is off the form. One more word under new X squared minus X squared. They and we know that integration off one over. And the route a square minus X squared is equal to sign in words x by eight. Let's see they So we will try to find Make it in the form off this toe. Evaluate in further on. Duh. The answer would be somebody off the state. Right. So let us find try to find the values off a and ex premier so you can see this is three square nine is three square. So it began. Have est three and here X is two weeks. This X is off the form two weeks. So let us say we take a capital X, which is equal to two weeks. They So this one over and the route three square minus two x holds quirk and Britain is one over in the route three square minus Capital X squared. And then we can further up like this dessert, right? So since we assumed exc EPT, Alexis is equal to two x and let's differentiate both sides So the X is equal to Cody X. OK, and now let's transform this expression from small extra capital X so this integral will be Let's say I this will be equal to not only X physical took the capital X. So this will be the capital X Over on the road three square minus two x whole square Just two weeks was equal to capital X. This means this is capital X square, right? So now we're gonna play this formula. So the formless is one over and root a square minus X squared and giggle is equal to sign inwards Expert A. So this will be signing worse Capital X by a three plus e they, with his equal do signing words two weeks extra capital exquisite will look to x from here . Right? So this means the answer is signing words two weeks by three. Let's see. So this is our final answer for the first example. No, let's take our second example. So our second example is we have to evaluate in giggle cost 66 times under route one plus signed six X and this is the X. So we have the Wellard in legal for this. So how we will go on took out like this. As you can see this function when plus sign six x, this function is is having its different differentiation in the question itself as different station of one plus sign. Success is six course xx. So I just ignoring the constant part. You can see different station off. This part is present in the question itself, right? So actually, we will be a playing here. It will be applying the integration by substitution. Okay, so let us assume one plus sign 66 is equal to Dean. Right? Are you can assume on the road one plus in six X is also esti. But since the friendship of this part is present, so we will assume this s t right differential of this part is 6466 So this is president, right? So now different Differentiating both sides. So this is zero plus cost 66 multiplied by different section of this part like the 66 X So this will be cost six x multiplied by six. So this we have done by Children this times DX is equal to Didi. Did this means six koszics course 66 The X is equal to DT or we can right this form in the form of course six XB x since course 6 60 excess president in the function. So this will be won by six DT so we can replace this caustic 60 x But this on the place one person's xx by this. So let us do that. So now we will apply the substitution off e in the above question. So by substituting one plus size six X is gonna be the integral Let's say I comes as so cost six x The X is won by 61 basics is a constant, so it will come out of the integration. Right? So beautiful remain Indian legal on the road one plus in six x will be in the room e right . So under duty comes in the indignant part. So are reserve will be Isaac will do one by six Integral Rudy, baby. Not this route Peak and Britain s when by 60 says this Rubik and readiness needed a powerhouse. Okay, Now you can apply the power rule here, do you? If I loading giggle set apart by powerful, this will be he to the power one by two. Less 1/1 by two less one. And, of course, plus sees there. So this will be on this will be three by two. So by reversing it, it will be to buy three. And this will be a to the power three by two. Let's see. So canceling this This this is one. This is three. So this will be won by nine. He to the power three by two. Let's see. And what was assumed Rusty this one plus signs XX So let's replace this one. Plus science xx again. Here s t to find the answer in the in the X itself. So the question was given In the terms of X, we have to give the answer in terms of X. So that's why I substituted back the value of P two sides when blessing six x right lies. So this is our solution to example to now let's move forward. You are heard. Example. Our third miscellaneous example is we have to evaluate. Let's say I is equal to in Eagle X to the power four over there. X minus one aims X squared plus one. You know what, like sort of function of this? This one? This is an international function late? No, which type of Russian function is is this is proper or improper Now, in proper, rational function, we have degree off the numerator less than the degree of denominator. As you can see, degree of denominator, rest three. But the degree off numerator this for which means they re off the numerator is greater than that degree off denominator, which means it's not a proper, rational functions. It is an improper aeration function. So we have to know in the National Function Video, we're discussing approach. We've learned the integral, but the approach was partial. Fraction become position and approach was applicable on Leone population functions. So in order it will load this integral by partially faction, become position matter. We have done this improper, rational function into a rash in function. Right? So let us do that. So how we will turn this improper, rational function into a rational function. We need to divide this numerator by this denominator. So after dividing what people up So I leave the vision part you guys, you guys can multiply this and value would be question my deriding Expect to the part for by the multiplication of this, right. So the answer comes like X plus one as the question. And the remaining part is one word. The multiplication of this is exc The denominators X Q minus X squared plus X minus one. Right. So this is the result after dividing this export with this? No, this is divided into two parts. This part is very easy to integrate this you can easy. Is Linda great? By applying the power will here on the needle of oneness obviously X. So this is fairly easy to integrate to the remaining parties. This and here This function is now a proper, rational function. That means you can apply a partial fraction the composition technique here in the function they so let us move forward. Who first you alluded in legal for this part. So this part can believe in us Van word. But this was X minus one times X squared plus one. Right, So this is one over X minus one times x squared plus one. And now, by the partial fraction the conclusion technique, we will assume this as a over the first factor. That is X minus one plus. Now, since the second factor, this cannot be fact raised further. So this is the this X squared plus one is a factor, right? So Xcor plus one is off the degree to that means in the numerator reveal. Assume a term off at this time we express. See, So this will be be explicit over X squared plus one right now, By taking the common denominator here, we will help you equations. Sorry, we will have three questions by quitting the coefficient off X Square X and the constant part. Right. So we will have three questions from here. Right? So we will have three questions to solve for a B and C. So I lived this part to you guys to Seoul for a B and C So when you saw for a B and C Finally you got a is equal to when my do b is equal. Toe minus one by two on C is equal Do again minus on my toe. Right. So these are the values when you try to solve those equations. So now this one over X minus one times X square plus one Can Britain is now a where X minus one. This was what one might do X minus one minus one might show. No, we explosive. So be X this over X squared plus one. Let's see. Over X squared plus one. So I've split. Did this dumb as well, right? So finally, what is left? Now we have successfully breaking broken This, uh, this thing. So this was X squared. This was explored this one day, so this finally can greatness. The first time was explicit one and the remaining comes us. The remaining terms are when? No. Where two X minus one minus one. Might do excellent X squared plus one minus when my two x squared plus one. So finally we have to evaluate the integral for this. These Fordham's Thank So the integral of this is very easy This is equal to X square by two for this X plus X And for this part, the constant barred one by do comes out off the integration part on the integral of one over X minus one Is Ellen more X minus one. And this indeed All here began a zoom minus one by toe as a constant. So this will be taken. All of the indications are integral will be DX over under the X, over one over X squared plus one. Right. So the integral will be one over X squared, plus one so integral of one over X squared plus minus. And in birds X by is one here. So this little bit an inverse Exxon. So finally, what is remaining is the giggle off this part? So in the middle of this part, this is integral off excellent X squared plus one like this on this part of the only too many? No. Here in this pipe, what really will be a plane? So as you can see the differential off, this different station off Xcor plus one is two X plus zero. That is two weeks, which is somewhat present in the numerator, so if we multiply this by two in the denominator and multi play with the Numa Numerator also too. So the differentiation of this part gets present here, Right? So now in this part we can apply the integration by a substitution. Okay, so by in traditional substitution, let us right the remaining which the terms of the terms of integration which we have calculator So that is X squared by Do this X minus when plus one by two Ellen more X minus one minus one. Way to eat And in what sex this have we have calculated. And the remaining part Waas It is minus one by four Indignity to X over. Expert, This one right, So minus. This is minus one by four in Diggle two weeks over. Expert less one late. So let us apply this substitution rule here. Right, So by substitution rule this will be a zoom dusty. That means it's differential. Look to x V x. This will be cool. Don't be so. This will be dp and this will be so integral off one by P baby. So that will be Ellen. More p right. So this will be minus one by four Ellen Morty. And what is eat is X squared plus one. So our final answer is X square by two plus X plus one bite Do Ellen More X minus one minus one by to an inverse X and the remaining park for From this this is minus one by four Ellen Morty. Morty is Ellen more X squared plus one. They and plus C off course, the concern off integration. So this is the final answer to this problem. So these were some of the miscellaneous examples which we have to work through this video. Thank you. 26. 25 Miscellaneous Examples - Part 2: Hi guys. In this with you, we will discuss some more miscellaneous examples on indefinite indignance. So let's start with our fourth example. Fourth example is we have to evaluate in giggle off one over under route under root X minus two the X So we happen value indeed all of this part. So as you can see distance and they do under the X minus two. So this time is ah, are done, which is under the radical sign like this is only heard, um which is under the radical side. So Daphnis this overall indignant is an irrational function. That means we have to make us substitution in order to make this fashion function transformed into a lesson function. So let's make a substitution. There's only one time which is under the illegal sign. So let us suppose you is, It will do ruethling load X minus two. Let's suppose this s u right. This means we can square both sides. So that means you square is going to on the X minus two. Or we can transform, which we can make this money store plus toe here. So that will be you squared plus two. Is it good toe little mix. And to find the value of X, we square it once again. So this will be you. Square place to hold square is equal to X which leads us to X is equal toe You been about food? Less so This is a square plus B squared. That is four plus two way be that is for use with right so excessive will do this they So now excess it will do. You did about four less for you square. Let's for right. So that's calculated the value of B Ex premier so different shipping both sites. So the differential of X dx will be Quito in our different station off this part that would be that will be find out by the someone. So first bled first to be different, distinct. But before you do less four times dispensation of use words that this to you. So when you read to you unless this is legal since different section of forest zero might be black back differential off you Right? So this is a similarly like if we have If we denigrate, be X If it delivered this part with respect to you So this will be the X by the you will be for you You bless. And you Let's go. So we're just so this Do you just get smelly player? So we're just different differentiating with respect to bury with itself. So I'm just writing different seal off access the X And here I am writing different tale of us to you. Right. So this is this. This is just a short time. So this will mean for you Q Bless you being So we have successfully completed the calculated the value of X and the value of BX. So let's come back to our original question. So let us suppose that integration indeed. All as I said that I was in Diggle off one word under your ex under root off Underwood ex Minister. They select a plan that I lose off X Indians in this integral. So this will be nor the X is equal to for the new cube less eight You, James B. And under the X minus two. This whole waas substituted as this was assumed. That's you right initially. So this is you Okay. Now you see, you can cancel out this you from denominator and the numerator, so this will be integral off for you. Square less aid and this is deep. So this is a simple so indicated off Well, you scripless eight. So we take out four as a constant on four as a possible come all of the indications. So the final indignation will be four times use Quite listo do right, and this is very simple to integrate. So this will be four times now by the power tool integration Off you square is a U Q by three on integration Off to is two times you since it just two times when people will come out of the integration. And indeed all of one issue. So this one week till you and obviously let's see will come as a concert of Indication. So the answer is for you. Q by three plus eight to you. Let's see now, since the original question was in terms off X, so we have to substitute the value of you again, transformed this answer in the in the form off X. So let's since so we'll substitute the value of you in this cancer. So this will be four by three. No, you were substituted as this on the road off under the X minus two. Great. So Ukyo will be this they Unless it and they took off under root X minus two. Let's see. So this is our final answer. But this was our first example. No, let's move on to are second example or visit or holds our fifth example. So our fifth example is let's say I physical, which we have evaluated. So I is equal toe in Diggle Sign ICS over sine X minus eight on this is Deeks with the spectacles we have integrate. So this is our question. And so we have to find this guy, right? It is equipped with this. So how we will be progressing. So we need to somehow make some adjustments in this numerator in order to make something out common from the denominator. Okay, so let's think of any identity. So if you, uh, careful you observed that this signings, if you value with this this sine X here can be written as sign off X minus a Let's see do. I didn't sign X minus. Now why? I have written like this. I wouldn't like this in orderto applied the identity Sign off a plus B. Since you must be aware of this identity that sign off a plus B, is it a little signing course? Be bless Coursey saying be so. I just subtracted and added a value in order to apply this identity. Now why? I'm applying this identity. I'm applying this identity since this will be signed X minus it cause a less course X minus a sign A And the first time this sign next ministry will get cancelled on it will leave us to a better indignant in orderto indicate. Right. So that will be a little simple to integrate. Right? So that will help. Thought of this. I didn't really play. So let us apply this identity first, which will make you look this question a little more simple. Simple, simple. Right. So let's apply. Deciding this will be signed. A Dr Stein X minus A. This is beat course. Hey, less course off X minus E. James. Sign a they or where? Sign of X medicine. Right. And this is multiplied by natives. Okay, so this means No. This was indeed a sign off X minus E course. A less course off X minus e signing, like in the noted. And this was doing everybody Sino X minus. No, these are totems. Custom is their second armistice in the first time. This sign next, minus e. Will you get cancer from the site? Next, medicine on in the second term. This will be course X men, ISI oversight expenses or discourse were saying that this court So this will be caught off , Excellency. So let's split out these both terms, right? So it will be in Diggle off since this sine x minus able be cancer. So this will be coursey less. This is sort of X men, etc. My replied Well signing. And this is our intake final indignity. So it is like you're more simple. We're indignant, this indignant rather than integrate this So we just applied this identity in order to make Doesn't didn't look like this. Right? So now we can apply this some rule here since there are times when David and begin indeed it using the simple So first of all, let's integrate this time not This is cost a No. We're integrating with respect to X. That means this this course off a will be treated as a concert, since it is not containing any damn affects. But this will be causing no course. They will come out of the integral on This will be integral. Be extended. Or you can write it when Wendy X less. Now the Harris signing signing is again a constant with respect to X So sign it will come out of the integration is that on in Diggle will be from here. This time it will be caught off x medicine. So this will be now in Dingle of minus X. So this will be X course A plus sign a times integral off poured off X minus eight BX No, as we have discussed in our previous videos, integral off cortex is equal. Do locks annex. Are you gonna easily prove this Using substitution allegedly so integral Off court off X minus Able be law off. All you can see. But the right is that it does. Ellen off. Ellen. More. Yes. Sign X minus it. They since integral of court access logs annex. It is a law. A little more sign. Next one, is it? They less see? So this is that sunset. The second legs at 3 50 example, actually, so let's move on. Do are six example. So our sixth example is I is equal to in Diggle Log X or I should write that in Diggle off clinics A bit better cited as long based e x. So we have very well would I We just integral off law based EBX longest e Expect this is clinical. So we have three weathered, integral off clinics. So how we will work so, you know, differential off clinics, right? So differential off different station off L. A nexus one works with respect to X but you don't know any identity as for now, in order to invigorate the next. So what would it be? Are identity or what will be our whole in orderto if I do this indeed in Dingle, which will be will be so here we can apply indignation by parts Now how we will apply integration by parts since in integration by parts we need to functions. But here, as you can see, there is only one function. But if you assume this Ellen excess Ln X multiplied by one No, I think it began. As you can see two guns in there What's this? Clinics? And second is this Well, they now this is I've just randomly selected this person second now, remember an indication by parts we decide the priority off the first function by the I later. So what is the island? Would I stand for investing in verse? Diplomatic functions else Ted stands for longer active function. A stand for algebraic. Expect algebraic functions P stand for trigonometry functions and east Thanks for exponential. They know what is one. The one we can die does back to the policy, right? So when can we treated us? And as a brick function they so which will be our first function here. Since the logarithmic function comes before the algebraic function, that means the logarithmic function will be treated like that first function. So we assume likely are functions right So fast function will be Ln x and the second function will be. But so now let's a play be integration by parts. So by integration by parts be no that the rule is first function that this eliminates multiplied by integral off second function that is one minus in Diggle off different station off first function are different section of dysfunction. Ln x is one backs multiplied by immigration off second function that is one on this hole is indicated that this is this DX is off this integral. Now let's move forward. So in Diggle of Oneness X, that means this is X Lennox minus indeed the love When by X times integral Open again here , Libbie X Right to this will be when my x times x dx So this ex gets cancelled with the sex and the remaining his integral index which is again equal to X. So the final answer So the final owns it is X Ellen eggs minus integral off one. This will be quick X Ln X minus X and obviously we will be adding the constant off indication. So this is our final answer. So, guys, these were some miscellaneous examples on indefinite integral. Thank you.