How do I evaluate $\int_0^{\pi/2}\int_0^2 x^2y\cos(xy^2)\,dx\,dy$?

Question

The question is $$\int_0^{\pi/2}\int_0^2 x^2y\cos(xy^2)\,dx\,dy$$

I have solved in this way but I am not getting the answer which is $\frac{-\pi}{16}$. Is this process of mine correct ?Here is the image

Answer 1

Your work is correct, but the answer is not $-\frac{\pi}{16}$ as mentioned by Dushyant. Integrating by parts we have

$$\int_{0}^{2}\frac{x}{2}\sin\left(\frac{\pi^2}{4}x\right)dx$$ $$=\left[-\frac{2x}{\pi^2}\cos\left(\frac{\pi^2}{4}x\right)\right]_{0}^{2}+\frac{2}{\pi^2}\int_{0}^{2}\cos\left(\frac{\pi^2}{4}x\right)dx$$ $$=-\frac{4}{\pi^2}\cos\left(\frac{\pi^2}{2}\right)+\frac{8}{\pi^4}\sin\left(\frac{\pi^2}{2}\right)$$