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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Definite Integration

NDA 2018 · Mathematics · Definite Integration

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

What is the value of ∫−π4π4(sin⁡x−tan⁡x)dx\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}(\sin x - \tan x) d x ?

Options

  1. A$−12+-\frac{1}{\sqrt{2}}+\operatorname{\ell n}\left(\frac{1}{\sqrt{2}}\right)$
  2. B12\frac{1}{\sqrt{2}}
  3. C0
  4. D2\sqrt{2}

Correct answer

C. 0

Step-by-step solution

∫−π4π4(sin⁡x−tan⁡x)dx\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}(\sin x-\tan x) d x Let f(x)=sin⁡x−tan⁡xf(x)=\sin x-\tan xf(−x)=sin⁡(−x)−tan⁡(−x)=−sin⁡x+tan⁡x=−(sin⁡x−tan⁡x)=−f(x)\begin{aligned} & f(-x)=\sin (-x)-\tan (-x) \\ & =-\sin x+\tan x=-(\sin x-\tan x)=-f(x) \end{aligned}$So,So,f(x)isanoddfunction.Weknowis an odd function. We know\int_{-a}^a f(x) \cdot d x=0,if, iff(x)$ is odd function.
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