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  4. The integral ∫ ((1-(1/√3))( cos x- sin x)/(1+(2)√3 sin 2 x)) d x is equal to

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Q. The integral $\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x$ is equal to

JEE MainJEE Main 2022Integrals

Solution:

$ I=\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x$
$\frac{\sqrt{3}}{2} \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(\frac{\sqrt{3}}{2}+\sin 2 x\right)} d x $
$ \int \frac{\left(\frac{\sqrt{3}}{2}-\frac{1}{2}\right)(\cos x-\sin x)}{\sin 60^{\circ}+\sin 2 x} d x$
$ \int \frac{\left(\frac{\sqrt{3}}{2} \cos x-\frac{1}{2} \cos x-\frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \sin x\right)}{2 \sin \left(x+\frac{\pi}{6}\right) \cos \left(x-\frac{\pi}{6}\right)} d x $
$\int \frac{\left(\cos \left(x-\frac{\pi}{6}\right)-\sin \left(x+\frac{\pi}{6}\right)\right)}{2 \sin \left(x+\frac{\pi}{6}\right) \cos \left(x-\frac{\pi}{6}\right)} d x $
$ \frac{1}{2}\left(\int \frac{d x}{\sin \left(x+\frac{\pi}{6}\right)}-\int \frac{d x}{\cos \left(x-\frac{\pi}{6}\right)}\right) $
$ \frac{1}{2} \ln \left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{12}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}\right| $