How do I show the relationship between $$I_n:=\int_{0}^{\pi}sin(x)^ndx$$ and $$I_n:=\frac{n-1}{n}I_{n-2}$$ for when $n \in \mathbb{N}$ and $n≥2$
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1Hint: $\sin(x)^n = \sin(x)^{n-1} \sin(x)$ and integrate by parts. – Dark Jun 14 '16 at 09:12
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1twice ${}{}{}{}$ – Jun 14 '16 at 09:14
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By the way, this must have been asked before. I only find this at the moment, but I guess a better search will make this question into a duplicate. – mickep Jun 14 '16 at 09:21
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1Have a look at http://www.vias.org/calculus/07_trigonometric_functions_05_03.html – Claude Leibovici Jun 14 '16 at 09:24
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this includes a solution (by Norbert) to your question http://math.stackexchange.com/questions/575737/a-sinn-x-integral – Math-fun Jun 14 '16 at 09:52
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I found this by using the search option, in particular see http://math.stackexchange.com/search?q=int_0%5E%7Bpi%7D+sin%5En+x – Math-fun Jun 14 '16 at 09:54