If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Integration by parts: ∫x²⋅𝑒ˣdx

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.E (LO)
,
FUN‑6.E.1 (EK)

## Video transcript

let's see if we can take the antiderivative of x squared times e to the X DX now the key is to recognize when you can at least attempt to use integration by parts and it might be a little bit obvious because this video is about integration by parts but the one way to the clue that integration by parts may be applicable is to say look I've got a function that's the product of two other functions in this case x squared and e to the X and integration by parts can be useful as if I can take the derivative of one of them and it becomes simpler and if I take the antiderivative of the other one it becomes no more complicated so in this case if I were to take the antiderivative of x squared it does become simpler and it becomes 2x and if I take the antiderivative of e to the X it doesn't become any more complicated so let's assign f of X to be equal to x squared and we want that one to be the one where if I take the derivative become simpler because I'm gonna have to take the derivative of f of X right over here and the integration by parts formula and let's assign G prime of X let's assign G prime of X to be equal to e to the X because later I'm gonna have to take its antiderivative and the antiderivative of e to the X is still just e to the X so let me write this down so we are assigning we are saying that f of X I'll do it right over here f of X is equal to x squared in which case f prime of X is going to be equal to 2x and I'm not worrying about its cot the constants right now we'll just add a constant at the end to make sure that our antiderivative is in the most general form and then G prime of X G prime of X is equal to e to the X which means its antiderivative G of X is still equal to e to the X and now we're ready to apply this the right hand side right over here so all this this thing right over here is going to be equal to f of X which is x squared and let me put it right underneath x squared times G of X which is e to the X which is e to the X minus - the let me do that in that yellow color I want to make the colors match up - the antiderivative of F prime of X well F prime of X is 2x to X times G of X G of X is e to the X G of X is e to the X DX e to the X DX so you might say Sal we're left with another another antiderivative another indefinite integral right over here how do we do how do we solve this one and as you might guess the key might be integration by parts again and we're making progress this right over here is a simpler expression than this notice we were able to reduce the degree of this x squared and now is just a 2x so let's actually and what we can do to simplify this a little bit since 2 is just a scaler it's a constant it's multiplying the function we can take that out of the integral sign so let's take it this way so let me rewrite it this way we can only do that with a constant that's multiplying the function so let me put the 2 right out here and so now what we're concerned about is finding the integral let me write it right over here the integral of X e to the X DX and now this is another integration by parts problem and so let's again let's again apply the same principles of integration by parts and what I take as derivatives going to get simpler well X is going to get simpler when I take its derivative so now for the purposes of integration by parts let's redefine f of X to be equal to just X and then we can still have G of X equaling we are G prime of X equaling e to the X and so in this case let me write it all down f of X is equal to X F prime of X is equal to 1 G prime of X G prime of X is equal to e to the X G of X is equal to just the antiderivative of this is equal to e to the X so let's apply integration by parts again so this is going to be equal to f of X times G of X now f of X is X f of X is X G of X is e to the X e to the X minus minus the antiderivative of F prime of X well that's just 1 times G of X e to the X is just 1 times e to the X 1 times e to the X e to the X DX D X and remember all I'm doing right now you might have lost track of things I'm just focused on this antiderivative that antiderivative is that antiderivative there if we can figure out what it is we can then substitute back into our original expression now you might appreciate integration by parts what does this right over here simplify to what is the antiderivative of 1 times e to the X DX or or what is the antiderivative 1 times e to the X well it's just the antiderivative of e to the X which is just e to the X so this simplifies to X eat it times e to the x times e to the X minus the antiderivative of e to the X which is just e to the X so minus e to the X and then we can take this and substitute it back this is the antiderivative of this so we can substitute it back up here to figure out the antiderivative of our original expression so the antiderivative of our original expression we're getting really close is going to be equal to and let me use it's going to be equal to I'm gonna use different colors so we can keep track of things it's going to be equal to x squared times e to the X x squared times e to the X minus 2 minus 2 times all of this business so minus 2 times well this antiderivative we just figured out is this minus 2 times X e to the X minus e to the X and if we want now is a good time to put our plus C and of course we can simplify this this is equal to x squared I like to keep the same colors this is equal to x squared e to the X distribute the negative two you get minus 2x e to the X plus 2 e to the X and then finally and then finally plus C and we're done we figured out the antiderivative of what looked like a kind of hairy looking expression using integration by parts twice
AP® is a registered trademark of the College Board, which has not reviewed this resource.