$\int_{0}^{\pi^2}\int_{x^{1/2}}^{\pi} \sin ( x/y)\, dy\,dx$

Question

I need to solve this integral and I don't know

Please, I can't change the order of integration.

I feel like it's impossible. $$\int_0^{\pi^2}\int_{x^{1/2}}^\pi\sin\frac xy\,dy\,dx$$

Answer 1

The variable $x$ can take values from $0$ to $\pi^2$. For each such $x$, $y$ can take values from $\sqrt x$ to $\pi$. So, $y$ can take any value from $0$ to $\pi$ and, for each such $y$, $x$ can take any value from $0$ to $y^2$. So, your integral is equal to$$\int_0^\pi\int_0^{y^2}\sin\left(\frac xy\right)\,\mathrm dx\,\mathrm dy,$$which is equal to$$\int_0^\pi y-y\cos(y)\,\mathrm dy=\frac{\pi^2}2+2.$$