Question

$\int \frac{\csc x}{(\csc x-\cot x)}dx$

Answer 1

$\int \frac{cosecx}{cosecx-\cot x}dx$

$=\int \frac{\cos ecx\left(\cos ecx+\cot x\right)}{\left(\cos ecx-\cot x\right)\left(\cos ecx+\cot x\right)}dx$

$=\int \frac{\cos ecx\left(\cos ecx+\cot x\right)}{\cos e{c}^{2}x-{\cot }^{2}x}dx$

$=\int \left(\cos e{c}^{2}x+\cos ecx\cot x\right)dx$

$=-\cot x-cosecx+c$