Original subtitles

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Today we're going to be talking about how to use integration by parts to evaluate an indefinite integral.

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And in this particular problem we've been given the integral of x cubed times natural log or Ellen of

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x x.

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Now we've already been told to use integration by parts to evaluate this integral.

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But if you're new to integrals you may not recognize that this is an integration by parts integral right

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away.

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The question you have to ask yourself is can you evaluate this integral as it is as it stands right

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now.

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And the answer is probably no.

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You're not quite sure how to evaluate this integral with just basic integration techniques.

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This isn't just a simple polynomial it's a little bit more complex.

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So you're going to have to manipulate the function a little bit before you can take its integral if

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you have to manipulate the function the first thing you want to try is use substitution if you can use

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substitution.

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That's the simplest way to manipulate an integral so that it's easier to evaluate.

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But in this case you substitution isn't really going to get us anywhere.

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Integration by parts should jump out at us as a good way to manipulate this integral because we have

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two components to our integral are two functions inside of our integral.

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One is x cubed here and the other one is natural log of X right here.

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Whenever we have two functions that are multiplied together inside of our integral we know that the

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integral is probably a good candidate for integration by parts because our integration by parts formula

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tells us that we have two functions that are multiplied together.

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One that we call you and the other one which we call DVH.

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Our only challenge when we use integration by parts is to identify which one will be you and which one

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will be DVH.

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Now when it comes to identifying you and DVH what I like to do is identify you first and then everything

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else in the integral just becomes divi by default and I've already picked you.

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When it comes to picking a value for you.

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A lot of people like to use what's called the least rule and the rule really become intuitive to you

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after you've had some progress doing this.

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But what the league rule tells you is that if you have a logarithmic function inside of your integral

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that's probably what you want to assign you to first.

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So if you have a logarithmic function like laga X or the natural log of X that's probably what you're

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going to want to use for you if you don't have a logarithmic function you go to the next letter and

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you look for an inverse trigonometric function.

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And if you have one of those you might want to use that.

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And then I think it's algebraic function triggered the metric function an exponential function in that

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particular order.

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So because we have the first letter the L we have a natural log of X here and are integral.

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That's probably what we want to use for you.

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The lead rule is not foolproof it's not perfect but it's a pretty good rule of thumb to go by.

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You should try to assign you to one of these functions in this order.

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So we're going to say that you is equal to natural log of X and that leaves everything else in integral

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x cubed times.

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Dyaks we're going to assign divi to x cubed d x remember that everything we have inside of our InterOil

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here has to be allocated either to you or to TV.

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And because we said that natural log of x is probably our candidate for you everything that's left over

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x cubed and DX has to go here to divi that DX will always go with DV.

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So we assigned you and DVH.

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Now at this point what we need to find are d u the derivative of u and v the integral of DVH.

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And that's why I wrote these here on two separate Rosing going to make a little mini table.

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But I like to have my original functions here.

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And V in my top row and my derivative functions d u and DV in my bottom row.

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We're going to start with you and take the derivative to get you.

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Then we're going to start with divi and take the integral to get V.

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These are the four values we're going to need to plug into our integration by parts formula.

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So taking the derivative of you I take the derivative of natural log of x which is 1 over X and because

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I'm taking the derivative here I have to then multiply by DX.

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I move over here to dvh.

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I want to take the integral of x cubed x y when I take the integral the DX will drop away.

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And the integral of x cubed is 1 fourth X to the four.

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So those are my four values that I'm going to be plugging into my integration by parts formula.

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I'm going to be plugging them in to the right hand side over here on plugging them in to this part right

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here.

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The left hand side of our integration by parts formula where it says the integral of you times the that

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matches up with our original integral the integral of x cubed Ellen of X because we said that Ellen

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X was you and that x cubed T X was DVH.

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So we're going to say the integral we're going to make substitutions here and say that the integral

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of cubed L.N. of x dx our original integral is equal to now using our formula you Times V.

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So we're going to multiply you and V here and when we combine these together we're going to get one

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fourth X to the four times natural log of X. Then according to our formula we're going to subtract the

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integral of V times d you will see here as one fourth X to the fore and D U is 1 over x dx.

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When we multiply these two values together if we look at it over here we're going to get one fourth

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X to the four.

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We're going to multiply that by 1 over x x when we do that.

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We're going to get X's to cancel here and to get this X to cancel this X the fourth will become x cubed.

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And so what we're going to have inside of our integral here is one fourth x cubed dx.

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Now before we evaluate this integral one thing we can do is move this one fourth right here in front

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of our integral because it's a constant coefficient on this x cube term.

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So what we're going to do is move that one forth there one fourth out in front of our integral and all

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we're left with now is the integral of x cubed dx and we can start to see why the integration by parts

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formula is so useful.

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We started with an integral for execute natural log of x dx which we didn't know how to evaluate.

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All we have left now is the integral of x cubed which is extremely easy for us to evaluate in fact we

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already found it here when we took the integral of divi which was exactly x cubed dx we already know

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that the integral is 1:48 X to the fore.

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So given that we're going to evaluate the integral in here we'll get one fourth X to the four times

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natural log of X Minus One fourth and here's where we have our integral.

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We know that the integral of x cubed is one fourth X to the four.

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And now we can't forget to add c to account for the constant of integration because we took this integral

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all we have to do at this point to get our final answer is simplify this as much as we can.

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One way that we can simplify it is by factoring out one fourth X to the four because we have a one fourth

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X to the four involved in both of our terms here.

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So we'll factor out one fourth X to the four.

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And what's left of our first term is just this natural log of X here.

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So we'll get natural log of X minus.

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We took out this whole one fourth X of the force.

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All we have left is this negative one force we get minus one fourth and then we add here.

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See we keep that C to account for the constant integration and that's it.

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That's our final answer.

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And that's how you use integration by parts to evaluate an indefinite integral.