Which is the justification for this indefinite integral relation?

Question

Why is the following indefinite integral equation correct:

$$ \int \frac{\cot(x)}{\sin^2(x)} dx= -\frac{1}{2}\cot^2(x) $$

What are the necessary steps?

Answer 1

let $cotx=\dfrac{cosx}{sinx}$ then let $u=sinx$ then $du=cosx dx$

Answer 2

$$\int\frac{\cot x}{\sin^2x}\,dx=\int\cot x\csc^2x\,dx$$ Recall that $$\frac{d}{dx}\cot x=-\csc^2x$$ so substituting $u=\cot x$ will work nicely, since this gives $-du=\csc^2x\,dx$. You then have $$\int\cot x\csc^2x\,dx=-\int u\,du$$ which gives you the result you're wondering about.

Note that an antiderivative in terms of $\cot x$ isn't the only possible one. I'm referring to the identity, $$\cot^2x=\csc^2x-1$$