I'd like to calculate $\int_{-1}^1(1-x^2)^n\ dx$, but I have no idea how. After searching through this website, I found out that we can transform the integral into Beta function, by using substitution. However, I'm having trouble finding the right substitution. I tried using $x = 1 - 2t$. Then the integral becomes $$-\frac{4^n}{2}\int_{3}^{-1}t^n(1-t)^n\ dt.$$ What's inside the integral is similar to Beta function. But the boundaries are not. It should have been $$\int_0^1t^n(1-t)^n\ dt.$$ Any idea what substitution should I choose?
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If $n\in\Bbb N$ you can use binomial expansion. – Tito Eliatron Jan 14 '21 at 18:09
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Did you try $x =\sin \theta$ (or $\cos \theta$)? – sudeep5221 Jan 14 '21 at 18:10
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1Does this answer your question? How to Integrate this function $\int(1-x^2)^ndx$ – Jan 14 '21 at 18:11
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Also see: https://math.stackexchange.com/questions/2074931/show-that-int-011-x2ndx-2n-over-2n1 – Jan 14 '21 at 18:11