JEE Main 2022 — Mathematics Calculus

2022mcqhard

The integral $\displaystyle \int \frac{(1-\frac{1}{\sqrt{3}})(\mathrm{cos}x-\mathrm{sin}x)}{(1+\frac{2}{\sqrt{3}}\mathrm{sin}2x)}dx$ is equal to

Official previous-year question

Held on 26 Jul 2022 · Verified 6 Jul 2026.

Options

A
$\frac{1}{2}{\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{12})}{(\frac{x}{2}+\frac{\pi }{6})}|+C$
B
${\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{6})}{(\frac{x}{2}+\frac{\pi }{3})}|+C$
C
$\frac{1}{2}{\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{6})}{(\frac{x}{2}+\frac{\pi }{3})}|+C$
D
$\frac{1}{2}{\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}-\frac{\pi }{12})}{\mathrm{tan}(\frac{x}{2}-\frac{\pi }{6})}|+C$

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Solution

Let $\displaystyle I=\int \frac{(1-\frac{1}{\sqrt{3}})(\mathrm{cos}x-\mathrm{sin}x)}{(1+\frac{2}{\sqrt{3}}\mathrm{sin}2x)}dx$

Multiplying by $\frac{\sqrt{3}}{2}$ in numerator and denominator, we get

$I=\int \frac{(\frac{\sqrt{3}}{2}-\frac{1}{2})(\mathrm{cos}x-\mathrm{sin}x)}{(\frac{\sqrt{3}}{2}+\mathrm{sin}2x)}dx$

$=\int \frac{(\frac{\sqrt{3}}{2}-\frac{1}{2})(\mathrm{cos}x-\mathrm{sin}x)}{\mathrm{sin}\frac{\pi }{3}+\mathrm{sin}2x}dx$

$=\int \frac{(\frac{\sqrt{3}}{2}\mathrm{cos}x-\frac{1}{2}\mathrm{cos}x-\frac{\sqrt{3}}{2}\mathrm{sin}x+\frac{1}{2}\mathrm{sin}x)}{2\mathrm{sin}(x+\frac{\pi }{6})\mathrm{cos}(x-\frac{\pi }{6})}dx$

$=\int \frac{(\mathrm{cos}(x-\frac{\pi }{6})-\mathrm{sin}(x+\frac{\pi }{6}))}{2\mathrm{sin}(x+\frac{\pi }{6})\mathrm{cos}(x-\frac{\pi }{6})}dx$

$=\frac{1}{2}(\int \frac{dx}{\mathrm{sin}(x+\frac{\pi }{6})}-\int \frac{dx}{\mathrm{cos}(x-\frac{\pi }{6})})$

$=\frac{1}{2}(\int \mathrm{cosec}(x+\frac{\pi }{6})dx-\int \mathrm{sec}(x-\frac{\pi }{6})dx)$

$=\frac{1}{2}(\mathrm{ln}|\mathrm{tan}\frac{x+\frac{\pi }{6}}{2}|)-\frac{1}{2}(\mathrm{ln}|\mathrm{tan}\frac{x-\frac{\pi }{6}}{2}+\frac{\pi }{4}|)+C$

$=\frac{1}{2}\mathrm{ln}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{12})}{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{6})}|+C$

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Options

$\frac{1}{2}{\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{12})}{(\frac{x}{2}+\frac{\pi }{6})}|+C$

${\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{6})}{(\frac{x}{2}+\frac{\pi }{3})}|+C$

$\frac{1}{2}{\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}+\frac{\pi }{6})}{(\frac{x}{2}+\frac{\pi }{3})}|+C$

$\frac{1}{2}{\mathrm{log}}_{e}|\frac{\mathrm{tan}(\frac{x}{2}-\frac{\pi }{12})}{\mathrm{tan}(\frac{x}{2}-\frac{\pi }{6})}|+C$