Mathematics - Definite Integration Question with Solution | TestHub

${\int }_0^{\pi /4}{\tan }^{n}x\left(1+{\tan }^{2}x\right)dx={\int }_0^{\pi /4}{\tan }^{n}x{\sec }^{2}xdx={\int }_0^1{t}^{n}dt$ where $t=\tan x$ $I_n+I_{n+2}=\frac{1}{n+1};\Rightarrow {\mathrm{Lim}}_{x\to \infty }n\left[l_n+I_{n+2}\right]={\mathrm{Lim}}_{x\to \infty }n\cdot \frac{1}{n+1}=\frac{n}{n+1}=\frac{n}{n\left(1+\frac{1}{n}\right)}=1$