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Question Number 123133 by Khalmohmmad last updated on 23/Nov/20

$$\int \sqrt{\sin x}dx=?$$

Commented by MJS_new last updated on 23/Nov/20

$$\mathrm{see}\mathrm{question}84607$$

Commented by Dwaipayan Shikari last updated on 23/Nov/20

$$\sqrt{sinx}=\sqrt{1-{msin}^2\theta }$$$$1-sinx={msin}^2\theta$$$${(sin\frac{x}{2}-cos\frac{x}{2})}^2={msin}^2\theta$$$$2\left({sin}^2(\frac{x}{2}-\frac{\pi }{4})\right)={msin}^2\theta$$$$\theta =(\frac{x}{2}-\frac{\pi }{4})m=2$$$$E(z\mid m)={\int }_0^z\sqrt{1-{msin}^{2}x}dx$$$$\int \sqrt{sinx}dx=2E(\frac{x}{2}-\frac{\pi }{4}\mid 2)$$

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