Prove that for all positive integers n > 1,
Integrating both sides,
Substituting x = 0,
Substituting x = -1,
What am I doing wrong?
You are right and the question is wrong. You can check this by testing the formula with n=2. In that case, the sum becomes
That agrees with your answer (with n=2), but not with the given answer 0.
My guess is that the person who set this question forgot the constant when doing the integration. Congratulations to you for remembering that this is necessary.