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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2022 · Mathematics · Indefinite Integration

What is ∫( x)^ -1 / 2 ( x)^ -3 / 2 d x equal to?

What is ∫(sin⁡x)−1/2(cos⁡x)−3/2dx\int(\sin x)^{-1 / 2}(\cos x)^{-3 / 2} d x equal to?

Options

  1. Atan⁡x+c\sqrt{\tan x}+c
  2. B2tan⁡x+c2 \sqrt{\tan x}+c
  3. Ccot⁡x+c\sqrt{\cot x}+c
  4. D2tan⁡x+c\sqrt{2 \tan x}+c

Correct answer

B. 2tan⁡x+c2 \sqrt{\tan x}+c

Step-by-step solution

Let I=∫(sin⁡x)−12(cos⁡x)−32dxI=\int(\sin x)^{-\frac{1}{2}}(\cos x)^{-\frac{3}{2}} d x=∫1(sin⁡x)12(cos⁡x)32dx=∫dxcos⁡2x(tan⁡x)12=∫sec⁡2xtan⁡xdx\begin{aligned} & =\int \frac{1}{(\sin x)^{\frac{1}{2}}(\cos x)^{\frac{3}{2}}} d x=\int \frac{d x}{\cos ^2 x(\tan x)^{\frac{1}{2}}} \\ & =\int \frac{\sec ^2 x}{\sqrt{\tan x}} d x\end{aligned}$\begin{aligned} & \text{ Let } \tan x=t^2 \Rightarrow \sec ^2 x d x=2 t d t \\ & =\int \frac{1}{t} \times 2 t d t=2 \int 1 d t=2 t+c \\ & =2 \sqrt{\tan x}+c\end{aligned}
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