Mathematics - Definite Integration Question with Solution | TestHub

$$\begin{array}{rlrl}& I={\int }_0^{\pi }(\pi -x)\left(\left({\sin }^2(\sin x)\right)+{\cos }^2(\cos x)\right)dx\Rightarrow 2I=2\pi {\int }_0^{\frac{\pi }{2}}\left({\sin }^2(\sin x)+{\cos }^2(\cos x)\right)dx& \Rightarrow I=\pi {\int }_0^{\frac{\pi }{2}}\left({\sin }^2(\sin x)\right)+{\cos }^2(\cos x)dx=\pi {\int }_0^{\frac{\pi }{2}}\left({\sin }^2(\cos x)+{\cos }^2(\sin x)\right)dx& \Rightarrow 2I=\pi {\int }_0^{\frac{\pi }{2}}2dx\Rightarrow I=\frac{{\pi }^2}{2}\end{array}$$