Q66 / 105 · JEE Main 2008 · Mathematics

The value of $\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$ is

  • A
    $x+\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c$
  • B
    $x-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$
  • Correct answer
    C
    $x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$
  • D
    $x-\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c$

Solution

$$ \begin{aligned} & \sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}=\sqrt{2} \int \frac{\sin \left(x-\frac{\pi}{4}+\frac{\pi}{4}\right) d x}{\sin \left(x-\frac{\pi}{4}\right)} \\ & =\sqrt{2} \int\left(\cos \frac{\pi}{4}+\cot \left(x-\frac{\pi}{4}\right) \sin \frac{\pi}{4}\right) d x \\ & =\int d x+\int \cot \left(x-\frac{\pi}{4}\right) d x \\ & =x+\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c . \end{aligned} $$

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

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