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Integration Techniques Overview · Acedra
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0:00
okay in this video we are going to do something a little bit different as you can see we have this intergrals on the board but we are not going to solve them instead, we'll just talk about what technique shall we use for this integrations well this right here is meant to be for Cal to students so of course you have to know the integration by parts tricks up use up and also partial fractions
0:20
and you know the trig integrals identities and all that and I have this file for you guys in fact we have a total of twelve integrals in this video you guys can download this that link go up in the description and of course it's free alright so if you guys like this just give me a like and share this video with your classmates your teachers your students I would be really grateful thank you so
0:42
much okay so let's get started with the first one the integral of natural log of X over X to the third power first of all your subs not going to work let's see I have pretty much two functions the first one is natural log of x and then the second one is 1 over X to a third power I know how to integrate 1 over X 4/3 power I know how to differentiate natural log of X so in this case that's go ahead - despite integration by parts and the truth is whenever you have the integral
1:13
of X to some power times natural log of X you can do them by integration by parts and of course you can set this up by the t I set up the DI method this is just integration by parts this is the saying that you this is the thing the TV right pan you know I have another video in the discussion on that but this is just the first step alright let me differentiate natural log of X because if you
1:40
want to integrate natural of X you have to use integration by parts again so don't and then I will be integrating 1 over X to the third power of course this is the same as X to the negative 3 so this is the neck this is the same extra negative 3 times natural log of X using the question are parts for this and then remember differentiating this you get 1 of X integrating this at 1 which is
2:03
going to be negative 2 and then divided by that new power so we end up with negative 1 over 2 x squared in the denominator and then remember the product of this bag you know it's the first part of the answer and then the Prada of this row you still have to put in the integral and when you do this times that in fact you can't agree that so that's the idea and now let's talk about the next
2:30
one the integral C 10 to the fourth power of X huh first of all you should know some famous ones as well if you have secant to the third power you have to do that by integration by parts when you have secant of fourth power this is actually much easier you can just do some trig identities so I already stuff for you I will actually take our two of those you can't so we have secant square
2:54
X right here and then times secant square X right here and a DX of course and then from here I can write secant square X in terms of tangent so this is going to give me the integral this is in fact the same as tangent square X plus one like this a ten times secant squared X DX after this what do we do yes we can just do it you stop right here so ready stuff for you guys you take you to be tangent to the first power X and then you see t you will be secant square X DX and then
3:35
you can take this integral to the new world and you finish it over the earth you can be a calculus finisher as well right now let's take a look of number three two x plus three over x squared minus five x plus four first of all the derivative of the bottom is 2x minus 5 it's nice that is the same as the top right unfortunately you stopped at some work this is the rational function on the bottom we can factor it so we'll just factor this and we can do partial fraction
4:07
and when you factor this you get X minus one times X minus four and as you can see we are two different linear factors that means we can just do the partial fractions by the cover-up method so I already start working for you guys this is the integral and then the first fraction has X minus 1 as its denominator and then the second one has X minus 4 as its denominator to figure this out this is just going to be a number you can put on a you would like for now and this is
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a B now to figure this out you go back to the original you cover the same denominator which is X minus 1 and you have to ask yourself how can you make X minus 1 to be equal to 0 where X has to be 1 then you plug in 1 into the rest of the X on the top let's do this on your head okay 2 times 1 is 2 plus 3 is 5 on the bottom 1 minus 3 is negative 3 so what does this mean we end up with 5 over negative 3 that's 4 a negative 5 over 3 all right now let's do this speed right here X
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minus 4 for this denominator go by there copy this up and you have to plug in X equal 4 here and here on the top 4 4 times 2 2 times 4 simply 8 plus 3 is 11 divided by 4 minus 1 which is 3 so 11 over Street so P is 11 over Street like this and then from there you can just integrate this fraction in that fraction and because this is just X to the first power you end up with the natural log right
5:49
how do you touch you now let's take a look right here the inter coop x squared times tangent of X plus third power what do we do take a look of the inside X cube when you differentiate X cube you get 3x squared we do have 2 X quit on the outside that can help us out so we can just do this pal you stop let u equal to X cube and then from there we get tu equal to 3x squared DX and if you want
6:16
to isolate X I think I'll be easier let's divide this on both sides we get yet equals tu over 3x squared put this in the you world and as you can see the xql cancel out you will have 1 stir factor and you have to know how to integrate tangent of U so let me just make a note right here for you guys you also have to remember that the integral of tangent and let me just put on you for this for
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you guys this right here is equal to the natural log absolute value of thicket of U and then plus V right you have to know this so that later on you can just use this and of course to integrate this you can write this as sine over cosine and then do another substitution but you use you already so you can just stop you and all that and you do have to once a factor but you know I'm now working at
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all that's your job right so you have to remember this so you can finish that now let's take a look at this we have the integral of 1 over 1 plus x squared and then 25 over 2 power well this is so intimidating but inside here we have two terms 1 plus x squared and we have to know that it's just like this look at that 1 plus tangent squared this right here it's equal to secant squared it just want her you can put this right here yes we'll just do a trick stop for this one so we're
7:50
you have a number square plus x squared will take X to be tangent you let this X to be tangent take X to be tangent and we take this to the theta world and you really have to do tricks up here because on the top is just a 1 if you have X you can just do a regular goes up right you can just let u equal to this but unfortunately we don't have that take X to be tangent theta and don't forget yet is equal to secant squared theta the data you have to remember to put this back for DX
8:26
and then you also have to know that when you have 1 plus tangent squared theta this gives you secant squared theta yes it's kind of weird because the derivative this is you can square and then 1 plus tangent squared also secant squared but this is a trick identity is the calculus derivative right here all right so now let's take a look at the integral of e to the square root of x this right
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here seems connect possible right but the truth is if we take this integral to the you world you will see open to do so that's made to happen I will first let u equal to this namely square root of x and then before you differentiate both sides well let me just square both sides so I can get x equals to u square because this way I can just look at this NT friendship both sides I get DX that's equal to 2 u tu and then I can just take this integral to the you world and then you will
9:25
see that we will have the integral of e this is my U now and then times the DX is to you do you like this and then as you can see here we have to you to the first power times e to the U and yes in this case you can just do di method namely integration by parts so you're going to do the di seta right here in this case we can actually you know finish the derivative so put on to you right
10:01
here and then we'll integrate e to the U and then differentiating to you you get to differentiating to you get zero so you stop right that's the first step when you see the zero in the D column and as you can see when you integrate e to the u you always get e to the U so this is pretty much it and then of course the product of the diagonals along with the sine front they will be the answer
10:24
however once you finish that be sure you go back to the X world and of course I would eat that to you guys and now let's take a look of the other questions right here okay number seven integrating sine square X well in this case as you can see we don't have cosine X to help us out so don't don't use up yet in this case we have to actually use the identity to reduce the power when we have
10:52
sine square X we know this right here it's equal to one half which is the constant multiple we can take the to the front of the integration and then the integral again this is equal to 1/2 times 1 minus cosine of 2x like this DX and then price you can just go ahead integrate integrating Y in the X world and also integrating this in the X world and right here you have to do it use up but I would
11:20
eat that you guys now number eight the integral of 1 over square root of x plus 1 minus square root of x hmm in this case I don't think you sub can help us out of two square roots I don't think you sub can work this out nice divorce he maybe worked I don't know but I know that maybe I can get rid of the square roots in the denominator just like the Google days right we can multiply out the bottom and the top by the conjugate so we can say bye-bye to the square root in the
11:53
denominator so let's do that let's multiply the bottom by square root of x plus 1 and change the minus to a plus and then you have the square root of x and of course do the same on the top square root of x plus 1 plus square root of x and the beauty of this is that when you multiply this and that remember we end up with this thing squared which is just X plus 1 and then minus this square
12:21
which is just X and you see X minus X is just gone and then of course we have the plus 1 and you have this over 1 of course the over one doesn't matter in another world we are just integrating square root of x plus 1 and then plus square root of x like this and then put on the DX right here and you can put parentheses right here be sure you just mean kick with this and maybe you can do a little bit use substitution but the inside here is just X to the first power plus 1 the
12:56
interruption of this is just 1 so if that u equal to X plus 1 you just get you know tu to be DX so you can just go for it right so I put down this right here for you guys on the exams do not put on Tata touch okay you actually do other work and then show me the final answer right anyway number nine oh my god this looks crazy we have e to the x over secant X but don't forget that secant X in
13:27
the denominator it's the same as what it's just the same as cosine X so in fact this integral is the same as integrating e to the x times cosine X and do we recognize how to integrate this sure we do how do we do it yes integration by parts and this right here it's a famous situation that you are going to get repeating situation right so let me just put this down here for you guys t and I I need two of them okay and in fact it doesn't really matter which one you pick to be
14:00
integrated they're equally easy you can integrate cosine you can integrate e to the X doesn't matter right but let me just go ahead and differentiate e to the X and then integrate cosine X and you may be say maybe you want to integrate e to the X it's easier that's up to you this is just as fine anyway you get e to the X e to the X to differentiate this and then integrating Cossacks first you get positive sine X and then integrate this again you can negative cosine X look at the
14:35
function part e to the X and then cosines this and that repeats right this right here repeat so remember you multiply this and that along with the sign in the front this is the first part of the answer and then this this and that that's the second part of your answer don't forget this right here when you multiply you still have an integral and then you're going to dry this down in your
15:01
line and you will see you how to add this on both sides and you all have to take care of that it's a repeating situation right okay now let's look at number 10 we have the integral 1 over 1 plus cosine X huh again I don't have thanks to help me out what do I do well this right here I don't like it why I like this one minus cosine squared because that's the same as sine square X right oh it's okay let me just go ahead and multiply the bottom and the top by the conjugates similar to
15:37
that it's gonna work on nicely let's do it like 1 minus cosine X on the bottom and also on the top and you will see that this is the same as the top is just this which is 1 minus cosine X and on the bottom when you multiply this out you get 1 minus cosine squared in other work we get sine squared X and now we have more things to work with this is pretty straightforward now because we can just
16:09
split the fraction since we don't have one turn the denominator first integral is the integral of 1 over sine square X and then perhaps the me close this and then I will put on the - and then the integral and then we have the cosine X over sine square and let me write this down as sine x times sine X for good purpose you'll see this right here you just have to focus on integrating cosecant
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square X and you have to know what the gravity reading wheel for this one right 1 over sine is cosecant 1 over sine square this so in fact this is pretty easy now but for this one look at I have this over that this is like saying 1 times cosine over sine X so you can say this is minus the integral 1 over sine is cosecant X and then cosine X over sine is cotangent X DX and again if you recognize this is the R of T of some function then you can integrate that but again I'll leave
17:21
this to you guys or right here you could have done use up as well that's fine but once you put this down this form you can just recognize this by knowing the roof give some function it's equal to this which is very nice okay two more alright and once again let me know it gets like this one not if you guys do maybe I produce part two like the other ones for you guys okay number you have an X minus 4 over x2 1/4 power minus one the rub key at the bottom it's not the same
17:54
as the top partial fractions then right so let's factor the denominator this right here do K this is the difference of two squares first X square square minus one squared so first you can factor this as x squared minus one times x squared plus one alright and then this is still factorable you can factor this to be X minus 1 times X plus 1 but this is no longer factorable you keep it this is a
18:23
irreducible quadratic alright and then the general form for this situation is that well first of all you have this for its denominator so I'll put down X plus 1 well the first one I put our negative so I put a minus here anyway this is the linear that means the top has to be just a constant and then next I have another linear factor this is different than that so I put down X plus 1 and it
18:53
has to be a constant on the top next one we have x squared plus 1 this is irreducible quadratic so I will write down x squared plus 1 by the top has to be one degree less than this name de has to be a linear now so I will put a see X plus D obesity or that and then of course you're going to find out the APCD and the inter gradient right so how deep that you guys okay for this one right here
19:29
don't you raise the X square on the top and just say hey I want to have the X not have now be so easy you can just do it without for this situation right but you do have to handle this X square here we have square root so this is now a partial fraction situation and technically this is just one term the square root her so you don't do the conjugate like that well in the square root we have 1 minus x squared it has two terms yes we can do tricks up for this and we have to know
20:02
that 1 minus sine square will give us cosine squared you can also use this 1 minus cosine square to be sine squared but you know just take this X to be sine theta right of course you have to change that to the theta world so like x equals let X equal to sine theta that's the deal and then don't forget to change the DX so I will have to differentiate a sample size so I will get DX to be that a variety of sine theta you are doing the derivative right here okay 9 to grow so from here
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to here it's just either rub it here you get cosine theta the theta and that's pretty much it but I think I'll work this off you guys have some space why not here we have the integral on the top I have sine right that's X and then u squared a so we have sine square theta on the bottom here we have square roots and then we have 1 minus sine square again so we have sine square theta here is a super common mistake people don't like to put on a tee whatever this right
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here if you don't put down anything that's like driving which are you sleepy are very dangerous so don't do that remember DX you did the hard work so put it down this right here is cosine theta the theta okay and now we're clean things up right here in the denominator I just focus on this square root part 1 minus square is cosine square theta biting the square root after you can saw the square the square root don't worry about the absolute value just focus cosine theta
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because you just want to focus on integration and check this out cosine theta and cosine theta cancel each other nice D and how can we integrate sine square theta yes we do this again right so I will leave the rest to you guys all right so this is it hopefully you guys all find this video to be helpful and if you can sign Cal to know best of luck to you guys have a wonderful semester note your integrations integrations are a lot of fun anyway this is it [Music]
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