I need someone REALLY good at math, please!!!!!?

Question

Hello, I have some multivariable problems that can be done fairly easily, but I really need help! Here is the problem:

Evaluate the integral:

int(0-->infinity) [ int(0-->infinity) 1/((1 + x^2 + y^2)^2) dxdy]

This is a double integral. Thanks very much in advance!

Answer 1

∫(0 to ∞) ∫(0 to ∞) 1/((1 + x^2 + y^2)^2) dxdy
Change to polar coordinates: x = r cos θ, y = r sin θ, r^2 = x^2 + y^2, dx dy = r dr dθ.
= ∫(0 to π/2) ∫(0 to ∞) 1/(1 + r^2)^2 r dr dθ
Substitute u = 1 + r^2, du = 2r dr. Also note θ is not in the integrand, so we get a constant multiple:
= (π/2 - 0) ∫(1 to ∞) 1/u^2 (du/2)
= π/2 (1/2) [-1/u][1 to ∞]
= π/4 (0 - (-1))
= π/4.