Expert: Paul Klarreich - 10/9/2009

Question
∫ x^n e^x dx= x^n e^x - n∫x^n-1 e^x dx    Prove:?

Then use the above to evaluate:
∫x^3 e^x dx

Answer
Questioner: Dolly
Country: United States
Category: Calculus
Private: No
Subject: Calculus. Integration by parts.
Question: ∫ x^n e^x dx= x^n e^x - n∫x^n-1 e^x dx    Prove:?

Then use the above to evaluate:
∫x^3 e^x dx
.............................................
Hi, Dolly,

This is a 'reduction formula', right?  And you will be reducing the x^n to x^n-1,  That means you pick   u = x^n, so you can differentiate and reduce the power.  That is your clue.

In ∫ x^n e^x dx,

pick   u = x^n, because it is easy to diff.   du = nx^n-1
     dv = e^x dx, because it is easy to integrate.  v = e^x


Set up your integration by parts:

uv = x^n e^x

-∫ v du = - ∫ n x^n-1 e^x dx

Put them together, and that should do it, I think.

As to the second part, see my other answer for the proper style.  I don't think you'll have any more trouble.