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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate : $\int \large\frac{1}{\Large\frac{\sqrt 3}{2} \sin x + \cos x }$$dx$

$(a)\;2 \log | \sec (x-\pi/6)+\tan (x-\pi/6) |+c \\(b)\;\frac{1}{2}\log | \sec (x-\pi/6)+\tan (x-\pi/6) |+c \\(c)\; 2 \log |\sec (x-\pi/6)-\tan (x-\pi/6) |+c \\ (d)\;\frac{1}{2} \log | \sec (x-\pi/6)-\tan (x-\pi/6) |+c$

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A)
$\int \large\frac{1}{\Large\frac{\sqrt 3}{2} \sin x + \cos x }$$dx$
=>$\int \large\frac{1}{2} \bigg[\frac{1}{\Large\frac{\sqrt 3}{2} \sin x +\frac{1}{2} \cos (x) }\bigg]$
=> $ \large\frac{1}{2} \int \large\frac{1}{\sin (x + \frac{\pi}{6})}$
=> $\large\frac{1}{2}$$ \int \sec (x-\large\frac{\pi}{6})$$dx$
$\large\frac{1}{2}$$\log | \sec (x-\pi/6)+\tan (x-\pi/6) |+c$
Hence b is the correct answer.
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