If $I_n = \int \sin^n x \, dx$,then $n I_n - (n-1) I_{n-2}$ equals

If $\int \frac{1 + \sqrt{\tan x}}{\sin 2x} dx = A \log \tan x + B \sqrt{\tan x} + C$ then $4A - B =$

Evaluate the integral: $\int \frac{2x+3}{\sqrt{3x^2-2x+1}} dx$

For $x > 0$,if $\int (\log x)^5 dx$ is equal to $x[A(\log x)^5 + B(\log x)^4 + C(\log x)^3 + D(\log x)^2 + E(\log x) + F] + \text{constant}$,then $A + B + C + D + E + F$ is equal to

If $\int \frac{x-\sin x}{1+\cos x} dx = x \tan \left(\frac{x}{2}\right) + p \log \left|\sec \left(\frac{x}{2}\right)\right| + C$,then $p$ is equal to

$\text{If } \int \frac{1}{\operatorname{cosec} x+\cos x} d x = \frac{1}{2 \sqrt{3}} \log |f(x)| - \int \frac{\cos x-\sin x}{2+\sin 2 x} d x + c, \text{ then at } x = \frac{\pi}{3}, |f(x)| = $