Hi I'm currently revising for a maths module that I am taking as part of my physics degree.
All was going well until I hit a dead-end with this integral, any ideas how to evaluate it?
$$ \int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} \frac{dx~dy}{(x^2 + y^2 + a^2)^2} $$
I tried substituting in 2D polar coordinates to get: $$ \int^{2 \pi}_{0} \int^{+\infty}_{0} \frac{r~dr~d\theta}{(r^2 + a^2)^2} $$ From here I thought of substituting in: $$ r = a~\tan(u) \Rightarrow dr = a~\sec^2(u)~du $$ This gives: $$ \int^{2 \pi}_{0} \int^{\frac{+\pi}{2}}_{\frac{-\pi}{2}} \frac{a ~\tan(u)~\sec^2(u) ~ ~du~d\theta}{a^4~\sec^4(u)} = \frac{1}{a^3}\int^{2 \pi}_{0} \int^{\frac{+\pi}{2}}_{\frac{-\pi}{2}} \frac{\tan(u) ~ ~du~d\theta}{\sec^2(u)} $$
From here I can't see where to go. Am I on the right track here or have I just overcomplicated this problem?
Thanks! Sean.
By evaluating the iterated integrals separately, the hard part is computing
$$\int_0^{\infty} \frac{r dr}{(r^2 + a^2)^2}$$
Susbstituting $u = r^2 + a^2$ leads to an elementary integral.