Double Integral $\iint\limits_D\frac{dx\,dy}{(x^2+y^2)^2}$ where $D=\{(x,y): x^2+y^2\le1,\space x+y\ge1\}$ Let $D=\{(x,y)\in \Bbb R^2 : x^2+y^2\le1,\space x+y\ge1\}$.  The integral to be calculated over $D$ is the following:
\begin{equation}
\iint_D \frac{dx\,dy}{(x^2+y^2)^2}
\end{equation}
I do not know how to approach the problem.  I have tried integrating the function in cartesian coordinates but it doesn't seem to work out. I have also tried the variable change $u=x^2+y^2$ and $v=x+y$ (with the associated jacobian transformation) and again I cannot obtain the result.

Transforming to polar coordinates $(\rho,\phi)$, we find that on $x+y=1$ we have $\rho =1/(\cos(\phi)+\sin(\phi))$.  Therefore, we can write
$$\begin{align}
\int_D \frac{1}{(x^2+y^2)^2}\,dx\,dy&=\int_0^{\pi/2}\int_{1/(\sin(\phi)+\cos(\phi))}^1 \frac{1}{\rho^3}\,d\rho\,d\phi\\\\
&=\frac12\int_0^{\pi/2}\sin(2\phi)\,d\phi\\\\
&=\frac12
\end{align}$$
