These are nice problems that can enlighten a few people who are not used to non-routine substitutions. People who are well acquainted with this, can take one integral at a time and state the substitution and show the solution step by step.

State the substitution and show the solution for the following integrals:

1) Prove

2) For .Prove that . Evaluate the integral for n = 1,2,3.

3) For . Prove

The aim of this thread was to let beginner-integral-lovers enjoy and learn something new. So feel free to skip a few obvious steps while writing the solution.

P.S: I think I will not be available from tomorrow to the end of this month