Change of variables Double integral I have 
$$\iint_A \frac{1}{(x^2+y^2)^2}\,dx\,dy.$$
$A$ is bounded by the conditions $x^2 + y^2 \leq 1$ and $x+y \geq 1$.
I initially thought to make the switch the polar coordinates, but the line $x+y=1$ is making it hard to find the limits of integration. 

If you rotate your domain by $45$ degrees your line becomes $y=\sqrt{2} /  2$ and the integrand is invariant, so you get (also use the simmetry with respect to $x=0$):
$$
\iint_A \frac{1}{(x^2+y^2)^2} = 2\int_0^{\sqrt 2/2}\,dx\int_{\sqrt 2/2}^{\sqrt{1-x^2}} \frac{dy}{(x^2+y^2)}
$$
