Mathematics - Definite Integration Question with Solution | TestHub

${\mathrm{I}}_{\mathrm{n}}=0+\mathrm{n}{\int }_0^{\infty }{\mathrm{e}}^{-\mathrm{x}}(\sin \mathrm{x}{)}^{\mathrm{n}-1}\cos \mathrm{xdx}$ (by parts)

$$={\int }_0^{\infty }-\left((\sin \mathrm{x}{)}^{\mathrm{n}}+(\mathrm{n}-1)\left(1-{\sin }^2\mathrm{x}\right)(\sin \mathrm{x}{)}^{\mathrm{n}-2}\right){\mathrm{e}}^{-\mathrm{x}}\mathrm{dx}=\frac{\mathrm{n}(\mathrm{n}-1)}{{\mathrm{n}}^2+1}{\mathrm{I}}_{\mathrm{n}-2}$$

$$\therefore {\mathrm{I}}_{\mathrm{n}}=\frac{\mathrm{n}(\mathrm{n}-1)}{{\mathrm{n}}^2+1}{\mathrm{I}}_{\mathrm{n}-2}$$