Question

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

Answer 1

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

$=\int \left(\frac{x{\cos }^{2}x/2}{x{\sin }^{2}x/2}\right)dx\{\cos x=2{\cos }^{2}x/2-1or\cos x=1-2{\sin }^{2}x/2\}$

$=\int \left(co{t}^{2}x/2\right)dx$

$=\int (cose{c}^{2}x/2-1)dx$

$=\int cose{c}^2\left(x/2\right)dx-\int dx$

Let $x/2=t$

So, $dx=2dt$

$=\int cose{c}^{2}t2dt-\int dx$

$=-2\cot \left(x/2\right)-x+C$