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 $\displaystyle \int_a^b \sqrt{(x-a)(x-b)}\, dx = \frac{\pi (b-a)^2}8$

2) For $\displaystyle n \in \mathbb{Z}^+ , a > |b|,$.Prove that $\displaystyle \int_0^{\pi} (a + b\cos x)^{-n} \, dx = (a^2 - b^2)^{-(n - \frac12)} \int_0^{\pi} (a - b\cos y)^{n-1} \, dy$. Evaluate the integral for n = 1,2,3.

3) For $\displaystyle m, n \in \mathbb{Z}^+$. Prove $\displaystyle \int_a^b (x-a)^m(b-x)^n \, dx = (b-a)^{m+n+1}\frac{m! n!}{(m+n+1)!}$

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