We are given a reduction formula for $\int \sin^n xdx$, namely $$\int \sin^nxdx = -\frac{1}{n}\sin^{n-1}x\cos x + \frac{n-1}{n}\int \sin^{n-2}xdx.$$ I know how to derive this reduction formula. How can we use this formula and mathematical induction to show that $$\int_0^{\pi/2}\sin^{2n+1}xdx = \frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdot\ldots\cdot\frac{2n}{2n+1}?$$ I am confused as to what the base case should be, and whether strong or "normal" induction would be a better approach.....?
2026-04-03 02:36:47.1775183807
Use of mathematical induction to compute $\int_0^{\pi/2} \sin^{2n+1}xdx$?
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