Question

Integrate cosecx dx

Answer 1

$I=\int \csc xdx$

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

Put $\csc x-\cot x=t$

$\Rightarrow (-\csc x\cot x+{\csc }^{x}x)dx=dt$

$\Rightarrow dx=\frac{dt}{-\csc x(\csc x-\cot x)}$

$\Rightarrow I=\int \frac{\csc x(\csc x-\cot x)}{\csc x-\cot x}.\frac{dt}{(\csc x-{\cot }^{2}x)\left(\csc x\right)}$

$=\int \frac{dt}{t}=\log \left|t\right|+C$

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

Hence, solved.