Question

Integrate cosecx /cosecx - cotx dx

Answer 1

please find the solution in the attached image

Answer 2

The integration for the given integral is

$\int \frac{\csc \left(x\right)}{\csc \left(x\right)-\cot \left(x\right)}dx=-\frac{1}{\tan \left(\frac{x}{2}\right)}+C$

Step-by-step explanation:

We have been given that

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

Apply the u- substitution

$u=\tan \left(\frac{x}{2}\right)\\\\du=\frac{1}{2}\sec^2\frac{x}{2}dx$

Thus, the integral becomes

$=\int \frac{1}{u^2}du$

$=\int \:u^{-2}du$

$\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1$

$=\frac{u^{-2+1}}{-2+1}$

Substitute back $u=\tan \left(\frac{x}{2}\right)$

$=\frac{\tan ^{-2+1}\left(\frac{x}{2}\right)}{-2+1}=-\frac{1}{\tan \left(\frac{x}{2}\right)}+C$

#Learn More:

Integrate the definite integral

https://brainly.in/question/4630073