Question

cotx/cosecx-cotx,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

In the attachment I have answered this problem.        The integrand is modified in such a way that it is suitable for integration.       See the attachment for detailed solution

Answer 2

we have to find the value of $\int{\frac{cotx}{cosecx-cotx}}\,dx$

$\int{\frac{cotx}{cosecx-cotx}}\,dx$

we know, cosec²x - cot²x = 1
or, (cosecx - cotx)(cosecx + cotx) = 1
or, (cosecx - cotx) = 1/(cosecx + cotx)

now, = $\int{\frac{cotx}{\frac{1}{cosecx+cotx}}}\,dx$

$=\int{cotx(cosecx+cotx)}\,dx$

$=\int{cotx.cosecx+cot^2x}\,dx$

$=\int{cotx.cosecx}\,dx+\int{(cosec^2-1)x}\,dx$

$=\int{cotx.cosecx}\,dx+\int{cosec^2x}\,dx-\int{dx}$

$=-cotx-cosecx-x+C$

$=-(cosecx+cotx) -x +C$