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JEE Main 2004
Exam
JEE Main
Year
2004
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2004 · Mathematics · Indefinite Integration

∫ (d x)/( x- x) is equal to

∫dxcos⁡x−sin⁡x\int \frac{d x}{\cos x-\sin x} is equal to

Options

  1. A12log⁡∣tan⁡(x2−π8)∣+C\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}-\frac{\pi}{8}\right)\right|+C
  2. B12log⁡∣cot⁡(x2)∣+C\frac{1}{\sqrt{2}} \log \left|\cot \left(\frac{x}{2}\right)\right|+C
  3. C12log⁡∣tan⁡(x2−3π8)∣+C\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}-\frac{3 \pi}{8}\right)\right|+C
  4. D12log⁡∣tan⁡(x2+3π8)∣+C\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}+\frac{3 \pi}{8}\right)\right|+C

Correct answer

D. 12log⁡∣tan⁡(x2+3π8)∣+C\frac{1}{\sqrt{2}} \log \left|\tan \left(\frac{x}{2}+\frac{3 \pi}{8}\right)\right|+C

Step-by-step solution

∫dxcos⁡x−sin⁡x=12∫1cos⁡(x+π4)dx=12∫sec⁡(x+π4)dx=12log⁡∣tan⁡(x2+3π8)∣+C\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
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