# Thread: help with integration

1. ## help with integration

I need help with the following integral:

$\displaystyle \int_{0}^{\infty}\frac{x}{(x+1)^{n+2}}$, where $\displaystyle n \in \mathbb{N}$

Do I need complex integration for this?

2. Put u = x + 1. Then x = u - 1 and du = dx:

$\displaystyle \int_0^{\infty} \frac{x }{(x+1)^{n+2}} dx = \int_1^{\infty} \frac{u-1}{u^{n+2}} du = \int_1^{\infty} \frac{1}{u^{n+1}} du - \int_1^{\infty} \frac{1}{u^{n+2}} du$.

Does this help?

3. I've solved it using that method and got $\displaystyle \frac{1}{n(n+1)}$, but it looks like I have to solve it by complex integration.

4. Complex integration is beyond my scope. Hopefully a more knowledgeable user will help you out soon. Sorry.

5. Originally Posted by trelee
I've solved it using that method and got $\displaystyle \frac{1}{n(n+1)}$, but it looks like I have to solve it by complex integration.

What have you tried and where are you stuck? Have you thought about:

1. What contour to use?

2. What function to use?