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Get Answers to all your Questions
What is the integration of \(\sqrt{\textrm{sin}\left(x\right)}\) ?
Answers (1)
We wish to evaluate the integral
$\displaystyle I=\int \sqrt{\sin x}\,dx$.
Put $\displaystyle \theta=\frac{\pi}{4}-\frac{x}{2}$. Then $x=\frac{\pi}{2}-2\theta$ and $dx=-2\,d\theta$.
Also
\[
\sin x=\sin\!\left(\frac{\pi}{2}-2\theta\right)=\cos 2\theta=1-2\sin^2\theta.
\]
Hence
\[
I=\int \sqrt{1-2\sin^2\theta}\;(-2)\,d\theta
=-2\int \sqrt{1-2\sin^2\theta}\;d\theta.
\]
The integrand is now in the standard form of the integrand of the incomplete elliptic integral of the second kind.
If we denote the incomplete elliptic integral of the second kind by
$\displaystyle E(\phi\,|\,m)=\int_0^{\phi}\sqrt{1-m\sin^2 t}\;dt$
(with parameter $m$), then
\[
\int \sqrt{1-2\sin^2\theta}\;d\theta = E(\theta\,|\,2)+C.
\]
Therefore substituting back $\theta=\dfrac{\pi}{4}-\dfrac{x}{2}$ we get
\[
\boxed{\,\displaystyle \int \sqrt{\sin x}\,dx
= -2\,E\!\left(\frac{\pi}{4}-\frac{x}{2}\,\Big|\,2\right) + C \,}
\]
where $E(\phi\,|\,m)$ is the incomplete elliptic integral of the second kind and $C$ is the constant of integration.
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