# How do you take take this integral?

1. Feb 22, 2009

### orthovector

how do you take take this integral?

$$\int_{0}^{r} x^2 e^{-2x} dx$$

2. Feb 22, 2009

### gabbagabbahey

Re: integral

Use integration by parts (twice)....the derivatives of $x^2$ are easy to find, and likewise for the antiderivative of $e^{-2x}dx$

3. Feb 22, 2009

### ice109

Re: integral

whats the upper limit? if it's infinity answer is 1/8

4. Feb 22, 2009

### orthovector

Re: integral

do you know how this integral turns into

$$\frac {N!}{a^{N + 1}}$$ if I take the integral from 0 to infinity? N = 2 and a = 2

5. Feb 22, 2009

### orthovector

Re: integral

6. Feb 22, 2009

### gabbagabbahey

Re: integral

If N=2 and a=2, then $$\frac {N!}{a^{N + 1}}=\frac {2!}{2^{2 + 1}}=\frac{1}{4}$$ which is what you should be getting using by parts.

Are you getting something different?

7. Feb 22, 2009

### orthovector

Re: integral

I was trying to derive the general expression

$$\int_{0}^{\infty} x^n e^{-ax} dx = \frac{n!}{a^{n+1}}$$

how is this so?

8. Feb 22, 2009

### gabbagabbahey

Re: integral

Use integration by parts n times and remember the definition of factorial; $n!=n(n-1)(n-2)\ldots (2)(1)$