Mathematics - Definite Integration Question with Solution | TestHub

$I_n={\int }_0^{\pi }{e}_{II}^x(\sin x{)}^{n}dx$

$$\begin{array}{rlrlr}& I_n=0-{\int }_0^{\pi }\underset{II}{\underbrace{n{e}^x}}\underset{I}{\underbrace{(\sin x{)}^{n-1}}}\cos xdx& & & \end{array}$$

$$I_n=0-n\left\{0-n{\int }_0^{\pi }{e}^x(n-1)(\sin x{)}^{n-2}{\cos }^{2}xdx+\int e^x(\sin x{)}^{n-1}\sin xdx\right\}$$

$$I_n(1+n+n(n-1))=n(n-1)I_{n-2}$$

$$I_3=\frac{3}{5}{I}_1$$