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$$\int \sqrt{\sin x} ~dx.$$

Does there exist a simple antiderivative of $\sqrt{\sin x}$? How do I integrate it?

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It does not have an elementary antiderivative. If calculating this integral is a step in a wider problem, it might be worth posting the wider problem, because it's likely that you've gone wrong somewhere in your working. –  Clive Newstead Aug 1 '12 at 18:23

1 Answer 1

up vote 18 down vote accepted

Since $\sqrt{\sin(x)} = \sqrt{1 - 2 \sin^2\left(\frac{\pi}{4} -\frac{x}{2}\right)}$, this matches with the elliptic integral of the second kind: $$\begin{align*} \int \sqrt{\sin(x)} \mathrm{d} x &\stackrel{u = \frac{\pi}{4}-\frac{x}{2}}{=} -2 \int \sqrt{1-2 \sin^2(u)} \mathrm{d} u\\ &= -2 E\left(u\mid 2\right) + c = -2 E\left(\frac{\pi}{4}-\frac{x}{2}\middle|\, 2\right) + c \end{align*}$$ where $c$ is an integration constant.

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I fixed your delimiters. Remember: comma for modulus, bar for parameter! –  J. M. Aug 2 '12 at 2:58

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