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Question Number 152608 by ZiYangLee last updated on 30/Aug/21

$$\mathrm{By}\mathrm{using}\mathrm{the}\mathrm{substitution}x=\cos 2\theta ,$$$$\mathrm{prove}\mathrm{that}\int \sqrt{\frac{1+x}{1-x}}dx=-\sin 2\theta -2\theta +C$$

Answered by Olaf_Thorendsen last updated on 30/Aug/21

$$\mathrm{F}\left(x\right)=\int \sqrt{\frac{1+x}{1-x}}dx$$$$\mathrm{F}\left(\theta \right)=\int \sqrt{\frac{1+\cos 2\theta }{1-\cos 2\theta }}(-2\sin 2\theta d\theta )$$$$\mathrm{F}\left(\theta \right)=\int \sqrt{\frac{{2\cos }^2\theta }{{2\sin }^2\theta }}(-4\sin \theta \cos \theta d\theta )$$$$\mathrm{F}\left(\theta \right)=\int \mid \cot \theta \mid (-4\sin \theta \cos \theta d\theta )$$$$\mathrm{F}\left(\theta \right)=-4\int {\cos }^2\theta d\theta )$$$$\mathrm{F}\left(\theta \right)=-2\int (1+\cos 2\theta )d\theta )$$$$\mathrm{F}\left(\theta \right)=-2(\theta +\frac{1}{2}\sin 2\theta )$$$$\mathrm{F}\left(\theta \right)=-\sin 2\theta -2\theta +\mathrm{C}$$

Commented by puissant last updated on 30/Aug/21

$$GenialMr!!!$$

Answered by Ar Brandon last updated on 30/Aug/21

$$I=\int \sqrt{\frac{1+x}{1-x}}dx,x=\cos 2\vartheta$$$$=-2\int \sqrt{\frac{1+\cos 2\vartheta }{1-\cos 2\vartheta }}\cdot \sin 2\vartheta d\vartheta$$$$=-2\int \frac{1+\cos 2\vartheta }{\sqrt{1-{\cos }^{2}2\vartheta }}\cdot \sin 2\vartheta d\vartheta$$$$=-2\int \frac{1+\cos 2\vartheta }{\mid \sin 2\vartheta \mid }\cdot \sin 2\vartheta d\vartheta$$$$=\mp 2\int (1+\cos 2\vartheta )d\vartheta$$$$=\mp (2\vartheta +\sin 2\vartheta )+C$$

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