Mathematics - Definite Integration Question with Solution | TestHub

$I_n={\int }_0^{\pi /2}(\sin x+\cos x{)}^{n-1}(\sin x-\cos x{)}^{\prime }dx$

$$\begin{array}{rl}& =2+(n-1){\int }_0^{\pi /2}(\sin x+\cos x{)}^{n-2}(\cos x-\sin x{)}^{2}dx\\ & =2+(n-1){\int }_0^{\pi /2}(\sin x+\cos x{)}^{n-2}\left[2-(\sin x+\cos x{)}^2\right]dx\\ & =2+2(n-1)I_{n-2}-(n-1)I_n\\ & \Rightarrow n{I}_n-2(n-1)I_{n-2}=2\end{array}$$