Mathematics - Definite Integration Question with Solution | TestHub

Consider $${\int }_0^{{\sin }^{2}x}{\sin }^{-1}(\sqrt{t})dt+{\int }_0^{{\cos }^{2}x}{\cos }^{-1}(\sqrt{t})dt$$ Let $\mathrm{I}=f(x)$ after integrating and putting the limits. $$\begin{array}{rl}& f^{\prime }(x)={\sin }^{-1}\sqrt{{\sin }^{2}x}(2\sin x\cos x)-0\\ & +{\cos }^{-1}\sqrt{{\cos }^{2}x}(-2\cos x\sin x)-0\\ & \therefore f^{\prime }(x)=0\Rightarrow f(x)=\mathrm{C}\text{ (constant) }\end{array}$$ Now, we find $f(x)$ at $x=\frac{\pi }{4}$ $$\begin{array}{rl}\therefore \mathrm{I}& ={\int }_0^{1/2}{\sin }^{-1}\sqrt{t}dt+{\int }_0^{1/2}{\cos }^{-1}\sqrt{t}dt\\ & ={\int }_0^{1/2}\left({\sin }^{-1}\sqrt{t}+{\cos }^{-1}\sqrt{t}\right)dt\\ & ={\int }_0^{1/2}\frac{\pi }{2}dt=\frac{\pi }{4}=\mathrm{C}\\ \therefore f(x)=\frac{\pi }{4}& \\ \therefore \text{ Required integration }=\frac{\pi }{4}& \end{array}$$