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Question Number 185985 by cortano1 last updated on 30/Jan/23

$$\int \frac{dx}{\sqrt{\sin x(1+\cos x)}}=?$$

Answered by ARUNG_Brandon_MBU last updated on 30/Jan/23

$$I=\int \frac{dx}{\sqrt{\sin x(1+\cos x)}}=\int \frac{dx}{2\sqrt{\sin \frac{x}{2}{\cos }^3\frac{x}{2}}}$$$$=\int \frac{\frac{1}{2}{\sec }^2\frac{x}{2}}{\sqrt{\tan \frac{x}{2}}}dx=\int \frac{d\left(\tan \frac{x}{2}\right)}{\sqrt{\tan \frac{x}{2}}}=2\sqrt{\tan \frac{x}{2}}+C$$

Answered by Gamil last updated on 31/Jan/23

Commented by ARUNG_Brandon_MBU last updated on 31/Jan/23

$$\frac{2}{\sqrt{\csc x+\cot x}}=\frac{2}{\sqrt{\frac{1+\cos x}{\sin x}}}$$$$=2\sqrt{\frac{2\sin \frac{x}{2}\cos \frac{x}{2}}{{2\cos }^2\frac{x}{2}}}=2\sqrt{\tan \frac{x}{2}}$$

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