Q176 / 270 · NDA 2011 (Phase 2) · Mathematics

What is $\int \frac{d x}{\sin ^2 x \cos ^2 x}$ equal to? where $c$ is the constant of integration.

  • A
    $\tan x+\cot x+c$
  • Correct answer
    B
    $\tan x-\cot x+c$
  • C
    $(\tan x+\cot x)^2+c$
  • D
    $(\tan x-\cot x)^2+c$

Solution

$\begin{aligned} \int \frac{d x}{\sin ^2 x \cos ^2 x} & =\frac{\int\left(\cos ^2 x+\sin ^2 x\right) d x}{\cos ^2 x \sin ^2 x} \\ & =\int \operatorname{cosec}{ }^2 d x+\int \sec ^2 x d x \\ & =-\cot x+\tan x+c \\ & =\tan x-\cot x+c .\end{aligned}$

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

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