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NDA 2011
Exam
NDA
Year
2011
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2011 · Mathematics · Indefinite Integration

What is ∫ (d x)/( ^2 x ^2 x) equal to? where c is the constant of...

What is ∫dxsin⁡2xcos⁡2x\int \frac{d x}{\sin ^2 x \cos ^2 x} equal to? where cc is the constant of integration.

Options

  1. Atan⁡x+cot⁡x+c\tan x+\cot x+c
  2. Btan⁡x−cot⁡x+c\tan x-\cot x+c
  3. C(tan⁡x+cot⁡x)2+c(\tan x+\cot x)^2+c
  4. D(tan⁡x−cot⁡x)2+c(\tan x-\cot x)^2+c

Correct answer

B. tan⁡x−cot⁡x+c\tan x-\cot x+c

Step-by-step solution

∫dxsin⁡2xcos⁡2x=∫(cos⁡2x+sin⁡2x)dxcos⁡2xsin⁡2x=∫cosec⁡2dx+∫sec⁡2xdx=−cot⁡x+tan⁡x+c=tan⁡x−cot⁡x+c.\begin{aligned} \int \frac{d x}{\sin ^2 x \cos ^2 x} & =\frac{\int\left(\cos ^2 x+\sin ^2 x\right) d x}{\cos ^2 x \sin ^2 x} \\ & =\int \operatorname{cosec}{ }^2 d x+\int \sec ^2 x d x \\ & =-\cot x+\tan x+c \\ & =\tan x-\cot x+c.\end{aligned}
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