Question

Solve:
$\int \frac{2\cos xdx}{(1-\sin x)(2-{\cos }^{2}x)}$

Answer 1

$\begin{array}{c}\int \frac{2\cos xdx}{\left(1-sinx\right)\left(2-{\cos }^{2}x\right)}\\ \int \frac{2\cos xdx}{\left(1-sinx\right)\left(1+si{n}^{2}x\right)}\sin x=t,\frac{dt}{dx}=\cos x\\ 2\int \frac{dx}{\left(1-t\right)\left(2-t^2\right)}\\ =2\left[\frac{1}{2}\int \frac{dt}{1-t}+\frac{1}{2}\int \frac{t+1}{t^2+1}dt\right]\\ =\int \frac{dt}{1-t}+\frac{1}{2}\int \frac{t+1}{t^2+1}dt\\ -\log \left|t-1\right|+{\tan }^{-1}t+\frac{1}{2}\log \left(t^2+1\right)+C\\ ={\tan }^{-1}t+\log \left(\frac{\sqrt{t^2+1}}{\left|t-1\right|}\right)+C\\ t=\sin x\end{array}$