Question

Evaluate: $\int \frac{cosx-cos2x}{1-cosx}dx$

Answer 1

$\int \frac{\cos x-\cos 2x}{1-\cos x}dx$

$\Rightarrow \int \frac{\cos x-2{\cos }^{2}x+1}{1-\cos x}dx$ $[\cos 2x=2{\cos }^{2}x-1]$

$\Rightarrow \int \frac{-(2{\cos }^{2}x-\cos x-1)}{(1-\cos x)}dx$

$\Rightarrow \int \frac{-(2{\cos }^{2}x-2\cos x+\cos x-1)}{(1-\cos x)}dx$

$\Rightarrow \int \frac{-\left[2\cos x\right[\cos x-1]+1[\cos x-1\left]\right]}{(1-\cos x)}dx$

$\Rightarrow \int \frac{(2\cos x+1)(\cos x-1)}{(\cos x-1)}dx$

$\Rightarrow 2\sin x+x+c$.