Integration Recurrence Formula

For Integration Recurrence Formula or reduction formula, it is important set a relationship between two consecutive terms by using mostly the integration by parts.

Question 1

For any integer $$m \ge 0$$ let $$\displaystyle I_m = \int_{0}^{1} x^m (x^2 – 1)^5 dx$$.
Prove that for $$\displaystyle m\ge 2, I_m = \frac{m-1}{m+11} I_{m-2}.$$

Question 2

Show that $$\displaystyle nA_n = \frac{2n-1}{2}A_{n-1}$$ for $$\displaystyle A_n = \int_{0}^{\frac{\pi}{2}} \cos^{2n}x dx, n \ge 1$$.