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Question Number 202167 by tri26112004 last updated on 22/Dec/23

$${\underset{0}{\int }}^1{\underset{x}{\int }}^{1}sin\left(y^2\right)dydx=¿$$

Answered by mnjuly1970 last updated on 22/Dec/23

$$answer:={sin}^2\left(\frac{1}{2}\right)$$

Commented by tri26112004 last updated on 22/Dec/23

$$Solution¿$$

Answered by witcher3 last updated on 22/Dec/23

$$\mathrm{x}⩽\mathrm{y}⩽10⩽\mathrm{x}⩽1\iff 0⩽\mathrm{x}⩽\mathrm{y}..\mathrm{\&}0⩽\mathrm{y}⩽1$$$${\int }_0^1{\int }_{\mathrm{x}}^1\sin \left({\mathrm{y}}^2\right)\mathrm{dydx}={\int }_0^1{\int }_0^{\mathrm{y}}\sin \left({\mathrm{y}}^2\right)\mathrm{dxdy}$$$$={\int }_0^1\mathrm{ysin}\left({\mathrm{y}}^2\right)=-\frac{1}{2}{\left[\cos \left({\mathrm{y}}^2\right)\right]}_0^1=\frac{1}{2}(1-\cos \left(1\right))$$$$\cos \left(1\right)=\cos (2.\frac{1}{2})=-{2\sin }^2\left(\frac{1}{2}\right)+1$$$$={\sin }^2\left(\frac{1}{2}\right)$$

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