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Question Number 205935 by EJJDJX last updated on 03/Apr/24

$$\int \underset{D}{\int }\left(4y^{2}sin\left(xy\right)\right)dxdy=???$$$$D:x=yx=0y=\sqrt{\frac{\pi }{2}}$$$$0⩽x⩽y0⩽y⩽\sqrt{\frac{\pi }{2}}$$

Answered by Berbere last updated on 03/Apr/24

$${\int }_0^{\sqrt{\frac{\pi }{2}}}{\int }_0^{y}4y^{2}sin\left(xy\right)dxdy$$$$={\int }_0^{\sqrt{\frac{\pi }{2}}}-4y{\left[cos\left(xy\right)\right]}_0^{y}dy$$$$={\int }_0^{\sqrt{\frac{\pi }{2}}}-4y[cos\left(y^2\right)-1)dy$$$$=-2{\int }_0^{\sqrt{\frac{\pi }{2}}}(cos\left(y^2\right)-1){dy}^2$$$$=2{\int }_0^{\sqrt{\frac{\pi }{2}}}(sin\left(y^2\right)-y^2)$$$$=2(1-\frac{\pi }{2})$$

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