Question

Evaluate: $\int \frac{1-\cot x}{1+\cot x}dx$

Answer 1

$I=\int \frac{1-\cot x}{1+\cot x}dx$

$\Rightarrow \int \frac{1-\frac{\cos x}{\sin x}}{1+\frac{\cos x}{\sin x}}dx$

$\Rightarrow \int \frac{\sin x-\cos x}{\sin x+\cos x}dx$

$\sin x+\cos x=t$

$(\cos x-\sin x)dx=dt$

$=\int -\frac{dt}{t}$

$=-ln|t|+c$

$=-ln|\sin x+\cos x|+c$.