Q112 / 160 · AP EAMCET 2022 (08 Jul Shift 1) · Mathematics

$\int \frac{d x}{\sin \left(x-\frac{\pi}{3}\right) \cos x}=$

  • Correct answer
    A
    $2 \log \left(\frac{\tan x-\sqrt{3}}{2}\right)+C$
  • B
    $2 \log \left(\sin \left(x-\frac{\pi}{3}\right) \cos x\right)+C$
  • C
    $2 \log (\tan x+\sqrt{3})+C$
  • D
    $2 \log (\sin x+\sqrt{3} \cos x)+C$

Solution

$I=\int \frac{d x}{\sin \left(x-\frac{\pi}{3}\right) \cos x}$ $=\int \frac{1}{\left(\frac{1}{2} \sin x-\frac{\sqrt{3}}{2} \cos x\right) \cos x} d x$ $\begin{aligned} & =2 \int \frac{\sec ^2 x}{\tan x-\sqrt{3}} d x \\ & =2 \log (\tan -\sqrt{3})+c \\ & =2 \log \left(\frac{n-\sqrt{3}}{2}\right)+c\end{aligned}$

Hence, the correct answer is option: 'A'.

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