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JEE Main 2008
Exam
JEE Main
Year
2008
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2008 · Mathematics · Indefinite Integration

The value of 2 ∫ ( x d x)/( (x-(π)/(4))) is

The value of 2∫sin⁡xdxsin⁡(x−π4)\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)} is

Options

  1. Ax+log⁡∣cos⁡(x−π4)∣+cx+\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c
  2. Bx−log⁡∣sin⁡(x−π4)∣+cx-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c
  3. Cx+log⁡∣sin⁡(x−π4)∣+cx+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c
  4. Dx−log⁡∣cos⁡(x−π4)∣+cx-\log \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c

Correct answer

C. x+log⁡∣sin⁡(x−π4)∣+cx+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c

Step-by-step 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}
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