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$y=\int \cos \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x$ is an equation of a family of
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parabolas
Let $I=\int \cos \left(2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x$
Put $x=\cos 2 \theta \Rightarrow d x=-2 \sin 2 \theta d \theta$
$\begin{aligned} \therefore I &=\int \cos \left(2 \tan ^{-1} \sqrt{\frac{1-\cos 2 \theta}{1+\cos 2 \theta}}\right\}(-2 \sin 2 \theta) d \theta \\ &=-2 \int \cos \left\{2 \tan ^{-1} \sqrt{\frac{2 \sin ^{2} \theta}{2 \cos ^{2} \theta}}\right\} \sin 2 \theta d \theta \\ &=-2 \int \cos \left(2 \tan ^{-1}(\tan \theta) \sin 2 \theta d \theta\right.\\ &=-2 \int \cos 2 \theta \sin 2 \theta d \theta \\ &=\int \cos 2 \theta d(\cos 2 \theta) \\ &=\frac{\cos ^{2} 2 \theta}{2}+C \\ &=\frac{x^{2}}{2}+C \end{aligned}$
Which is an equation of parabola.
Put $x=\cos 2 \theta \Rightarrow d x=-2 \sin 2 \theta d \theta$
$\begin{aligned} \therefore I &=\int \cos \left(2 \tan ^{-1} \sqrt{\frac{1-\cos 2 \theta}{1+\cos 2 \theta}}\right\}(-2 \sin 2 \theta) d \theta \\ &=-2 \int \cos \left\{2 \tan ^{-1} \sqrt{\frac{2 \sin ^{2} \theta}{2 \cos ^{2} \theta}}\right\} \sin 2 \theta d \theta \\ &=-2 \int \cos \left(2 \tan ^{-1}(\tan \theta) \sin 2 \theta d \theta\right.\\ &=-2 \int \cos 2 \theta \sin 2 \theta d \theta \\ &=\int \cos 2 \theta d(\cos 2 \theta) \\ &=\frac{\cos ^{2} 2 \theta}{2}+C \\ &=\frac{x^{2}}{2}+C \end{aligned}$
Which is an equation of parabola.
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