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.

Worked Example of Integration Recurrence Formula

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 \( \displaystyle A_n = \int_{0}^{\frac{\pi}{2}} \cos^{2n}x dx, n \ge 1\).


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