Question

Solve
$\int \csc xdx$

Answer 1

$\int cscxdx$

$=\int cscx\frac{cscx+\cot x}{cscx+\cot x}\cdot dx$

$u=cscx+\cot x$

$du=-cscx\cot x-{\csc }^{2}x$

$=\int \frac{cs{c}^{2}x+cscx\cot x}{cscx+\cot x}\cdot dx$

$=-\int \frac{cs{c}^{2}x+cscx\cot x}{cscx+\cot x}\cdot dx$

$=-\int \frac{du}{u}=-\int \frac{1}{u}du$

$=-log|u|+c$

$=-log|csc+\cot x|+c$.