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  4. If ∫ √ sec 2 x-1 d x=α log e| cos 2 x+β+√ cos 2 x(1+ cos (1/β) x)|+ constant, then β-α is equal to

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Q. If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to_______

JEE MainJEE Main 2023Integrals

Solution:

$ \int \sqrt{\sec 2 x-1} d x=\int \sqrt{\frac{1-\cos 2 x}{\cos 2 x}} d x $
$ =\sqrt{2} \int \frac{\sin x}{\sqrt{2 \cos ^2 x-1}} d x$
$ \text { put } \cos x=t \Rightarrow-\sin x d x=d t $
$ =-\sqrt{2} \int \frac{ dt }{\sqrt{2 t^2-1}} $
$=-\ln |\sqrt{2} \cos x+\sqrt{\cos 2 x}|+c $
$ =-\frac{1}{2} \ln \left|2 \cos { }^2 x+\cos 2 x+2 \sqrt{\cos 2 x} \cdot \sqrt{2} \cos x\right|+c $
$ =-\frac{1}{2} \ln \left|\cos 2 x+\frac{1}{2}+\sqrt{\cos 2 x} \cdot \sqrt{1+\cos 2 x}\right|+c $
$ \because \beta=\frac{1}{2}, \alpha=-\frac{1}{2} $
$\Rightarrow \beta-\alpha=1$