Question

Evaluate $\int \frac{1+x\sin x+\cos x}{x\left(1+\cos x\right)}dx$

Answer 1

$\int \frac{1+x\sin x+\cos x}{x\left(1+\cos x\right)}dx$

$=\int \frac{1+\cos x}{x\left(1+\cos x\right)}dx+\int \frac{x\sin x}{x\left(1+\cos x\right)}dx$

$=\int \frac{1}{x}dx+\int \frac{\sin x}{1+\cos x}dx$ (It $\cos =\theta$

$=\log x+\int \frac{dt}{\theta }$ $(-\sin xdx=dt$)

$=\log x-\log x+c$

$=\log x-\log \left(1+\cos x\right)+c$