Whenever you try to find the "circumference" of an ellipse you end up with an integral that looks like this: $$4a\int_{0}^{\pi/2}\sqrt{1-e^2\sin^2{\theta}} d\theta, $$ I know that there isn't an analytical solution to this because I've been told so, but how does one go abiut proving that statement?
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Usually Liouville's theorem is used. It's well known that this particular integral does not have an elementary solution. – Vasili Oct 24 '19 at 14:46
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You may be interested in this paper: http://math.stanford.edu/~conrad/papers/elemint.pdf – Vasili Oct 24 '19 at 14:52
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See also https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral and https://math.stackexchange.com/questions/15750/when-is-an-elliptic-integral-expressible-in-terms-of-elementary-functions – lhf Oct 24 '19 at 15:12