Q112 / 225 · JEE Main 2004 · Mathematics

$\int \frac{d x}{\cos x-\sin x}$ is equal to

  • A
    $\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}-\frac{\pi}{8}\right)\right|+C$
  • B
    $\frac{1}{\sqrt{2}} \log \left|\cot \left(\frac{x}{2}\right)\right|+C$
  • C
    $\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}-\frac{3 \pi}{8}\right)\right|+C$
  • Correct answer
    D
    $\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}+\frac{3 \pi}{8}\right)\right|+C$

Solution

$\int \frac{d x}{\cos x-\sin x}=\frac{1}{\sqrt{2}} \int \frac{1}{\cos \left(x+\frac{\pi}{4}\right)} d x=\frac{1}{\sqrt{2}} \int \sec \left(x+\frac{\pi}{4}\right) d x=\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}+\frac{3 \pi}{8}\right)\right|+C$

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

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