Question

What is the antiderivative of `csc^2x`?

Answer 1

The antiderivative of `csc^2x` is `-cotx+C`.

Why?
Before trying anything 'fancy' (subsitutuion, parts, trig sub, misc sub, partial fractions, et c.) try 'staightaway' antidifferentiation.

Do you know a finction whose derivative is `csc^2x`?

Go through the list:
`d/(dx)(sinx)=cosx`
`d/(dx)(cosx)=-sinx`
`d/(dx)(tanx)=sec^2x`

Hang on! that's good! The derivative if a `co` function has a minus sign and cofunctions, so `d/(dx)(cotx)=-csc^2x`

So, no, I don't know a function whose derivative is `csc^2x`, but I do know one whose derivative is `-csc^2c`. But this reminds me that:

`d/(dx)(-cotx)=-(-csc^2x)=csc^2x`

Therefore the antiderivative of `csc^2x` is `-cotx+C`