Try it out. Take the derivative of . Show us your work that leads you to believe the question is wrong.
A question says this: Show by differentiation that ln(csc x - cot x) is an antiderivative of csc x. I think the integral of csc x is -ln(abs(csc x - cot x)). I don't know why the sign appears to be different.
Are you familiar with the Fundamental Theorem of Calculus? That theorem proves that the operation of integration and the operation of differentiation ARE opposites. Any two distinct antiderivatives of a function differ by a constant (so the indefinite integral is any antiderivative of the function + an arbitrary constant).
But, if you still need convincing, consider this:
Let . Taking derivatives of both sides, we get