I'm trying to integrate $$\int_D y\cos^4x \, \mathrm dx \mathrm dy$$ where $D=\{(x,y)\in R^2;x^2+y^2<\pi\}.$ I'm thinking using the polar coordinates but doing so the integral would become,
$$\int_D \rho \sin\theta\cos^4(\rho \cos\theta)\ \, \mathrm d\theta \mathrm d\rho$$
and $D=\{(\rho,\theta)\in R^2;\rho^2<\pi,0\leq \theta \leq 2\pi\}$ which is even more difficult. Can someone explain me how to solve the exercise?
The professor says that this should have a quick resolution, so probably there is something I'm missing, please help me