# Homework Help: Volume integral

1. Nov 18, 2007

### jdstokes

1. The problem statement, all variables and given/known data

I'm trying to understand the solution to this integral

$\int_{\mathbb{R}^3} \frac{e^{i \mathbf{x} \cdot \mathbf{a}}}{\sqrt{r^2+1}}d\mathbf{x}$

where $d\mathbf{x} =dxdydz, r = \sqrt{x^2+y^2+z^2},\mathbf{a}\in \mathbb{R}^3$

3. The attempt at a solution

$\int_{\mathbb{R}^3} \frac{e^{i \mathbf{x} \cdot \mathbf{a}}}{\sqrt{r^2+1}}d\mathbf{x} = 2\pi \int_0^{\infty}\frac{1}{\sqrt{r^2 +1}}\frac{e^{iua}-e^{-iua}}{iua}u^2du$

Could anyone please explain to me how this first step was obtained?

Last edited: Nov 18, 2007
2. Nov 18, 2007

### nrqed

Just write $\vec{x} \cdot \vec{a} = x a cos \theta$ and the volume element as $d^3 x =x^2 dx d\phi d \cos \theta$ and integrate over cos theta from -1 to 1.