If $\int \frac{x^2\left(\sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=\frac{-x^2}{x \tan x+1}+f(x)+C$, then $$ f(x)= $$

7. If $\int \sin (101 x)(\sin x)^{99} d x$ $=\frac{\sin (100 x)(\sin x)^\lambda}{\mu}+C$ then, $\frac{\lambda}{\mu}=$

If $\int e^x\left(\sin ^2 2 x-8 \cos 4 x\right) d x=e^x f(x)+C$, then $f\left(\frac{\pi}{4}\right)=$

If $n$ is a positive integer greater than 1 and $I_n=\int \frac{\sin n x}{\sin x} d x$, then $I_{n+1}-I_{n-1}=$