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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Definite Integration

NDA 2010 · Mathematics · Definite Integration

What is the value of ∫_ π / 6 ^ π / 4 (d x)/( x x) ?

What is the value of ∫π/6π/4dxsin⁡xcos⁡x?\int_{\pi / 6}^{\pi / 4} \frac{d x}{\sin x \cos x} ?

Options

  1. A2ln⁡32 \ln \sqrt{3}
  2. Bln⁡3\ln \sqrt{3}
  3. C2ln⁡32 \ln 3
  4. D4ln⁡34 \ln 3

Correct answer

B. ln⁡3\ln \sqrt{3}

Step-by-step solution

∫π/6π/4−dxsin⁡x⋅cos⁡x=∫π/6π/4sec⁡2xtan⁡xdx=∫131dzz=[log⁡z]131=[log⁡1−(log⁡1−log⁡3)]=log⁡3\begin{aligned} & \int_{\pi / 6}^{\pi / 4} \frac{-d x}{\sin x \cdot \cos x}=\int_{\pi / 6}^{\pi / 4} \frac{\sec ^2 x}{\tan x} d x \\ & =\int_{\frac{1}{\sqrt{3}}}^1 \frac{d z}{z}=[\log z]_{\frac{1}{\sqrt{3}}}^1 \\ & =[\log 1-(\log 1-\log \sqrt{3})]=\log \sqrt{3}\end{aligned}
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