Question

Show that $\int cosecxdx=log|cosecx-cotx|+c$

Answer 1

$\int \csc xdx$

$=\int \csc x\left(\frac{\csc x-\cot x}{cscx-\cot x}\right)dx$

$=\int \left(\frac{{\csc }^{2}x-\csc x\cot x}{\csc x-\cot x}\right)dx$

Let $u=\csc x-\cot x$

$\Rightarrow du=(-\csc x\cot x-(-{\csc }^{2}x))dx$

$\Rightarrow ({\csc }^{2}x-\csc x\cot x)dx=du$

$\therefore \int \csc xdx$

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

$=\log \left|u\right|+C$

As $u=\csc x-\cot x$

$\therefore \int \csc xdx=\log |\csc x-\cot x|+C$

Hence proved.