Evalúe la integral doble iterada

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Hola Jhoana Sm!

$$\begin{align}&\int_{\frac{\pi} 2}^{\pi} \Bigg[ \int_0^{y^2} sen \frac x y\ dx\ \Bigg]\ dy=\\&\\&\int_{\frac{\pi} 2}^{\pi} \Bigg[-y \cos \frac x y \Bigg]_{x=0}^{x=y^2}\ \ dy=\\&\\&\int_{\frac{\pi} 2}^{\pi} \Bigg(-ycos \frac{y^2} y+ ycos 0 \Bigg)dy=\\&\\&\int_{\frac{\pi} 2}^{\pi} \Bigg (-y \cos y +y \Bigg)dy=\\&\\& \Bigg[-ysin y - cosy + \frac{y^2} 2 \Bigg]_{ \frac {\pi} 2}^{\pi}=\\&\\&0-(-1)+ \frac{\pi^2} 2 - \Bigg(- \frac {\pi} 2-0+ \frac{\pi^2}8 \Bigg)=\\&\\&1+ \frac 3 8 \pi^2+ \frac{\pi} 2\\&\\&\end{align}$$

Saludos

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