Solving $\iint \frac{1}{(x^2+y^2+1)^{3/2}} dx dy $

Question

This is a question in a book of statistics and probability. To prove that this function is a Probability density function, we should solve it to get the answer equals to 1. I haven't had to deal with these kinds of integrals for a while. I need a level by level solution with complete description.

Any help would be greatly appreciated.

problem

Answer 1

$$\iint_{\mathbb{R}^2}\frac{dx\,dy}{(1+x^2+y^2)^{3/2}}=\int_{0}^{+\infty}\pi\frac{2 \rho}{(1+\rho^2)^{3/2}}\,d\rho=\pi\left[-\frac{2}{\sqrt{1+\rho^2}}\right]_{0}^{+\infty}=\color{red}{2\pi}. $$