Let $f:[-\sqrt{2}\pi, \sqrt{2}\pi] \rightarrow \mathbb{R}$ a continuous function. Calculate $$\int_{-\sqrt{2}\pi}^{\sqrt{2}\pi} \int_{-1}^{1} f(y^{2})e^{\frac{|x|-y}{2}} \sin(x) dydx$$
Do you have any ideas for this problem? I have problems with this part, $$\int_{-1}^{1} f(y^{2}) e^{-\frac{y}{2}}dy$$
I think it should be integration for parts, with $u=e^{-\frac{y}{2}}$ and $dv=f(y^{2})$ since $f$ is continuous but then $v= \int_{-1}^{1} f(y^{2}) dy$?