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How do we go about computing the above integral? I figured we could write it as follows \begin{equation} \int_{0}^{\sqrt\frac{\pi}{2}}\int_{0}^{\sin(x^2)}\sin(y^2) \, \,\mathrm{d}y \, \,\mathrm{d}x \end{equation} But I really doubt that'd help in the first place. And am also not familiar with complex analysis :/. Would appreciate any assistance.

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    $\begingroup$ Have you tried changing the order of integration from $dydx$ to $dxdy$, and adjusting the bounds accordingly? As a reminder, one way that this can be done is to begin by sketching the region of integration. $\endgroup$ Mar 3 '19 at 14:45
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by chainging the order : $\int_{y=0}^{y=\sqrt{\frac{\pi}{2}}} \int_{x=0}^{x=y} \sin(y^2)~dx~dy = \int y\sin(y^2)dy = \frac{-\cos(y^2)}{2}|_{0}^{\sqrt{\frac{\pi}{2}}} = \frac{1}{2}$

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  • $\begingroup$ Could you explain how you adjusted your boundaries there? I don't see how $x$ runs from $0$ to $y$ $\endgroup$ Mar 3 '19 at 14:58
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    $\begingroup$ Oh wait nevermind I think I got it. $\endgroup$ Mar 3 '19 at 15:02

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