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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2010 · Mathematics · Indefinite Integration

What is ∫ ^n x x d x equal to? where c is a constant of integration.

What is ∫sec⁡nxtan⁡xdx\int \sec ^n x \tan x d x equal to? where cc is a constant of integration.

Options

  1. Asec⁡nxn+c\frac{\sec ^n x}{n}+c
  2. Bsec⁡n−1xn−1+c\frac{\sec ^{n-1} x}{n-1}+c
  3. Ctan⁡nxn+c\frac{\tan ^n x}{n}+c
  4. Dtan⁡n−1xn−1+c\frac{\tan ^{n-1} x}{n-1}+c

Correct answer

A. sec⁡nxn+c\frac{\sec ^n x}{n}+c

Step-by-step solution

∫sec⁡nxtan⁡xdx=∫sec⁡n−1x⋅sec⁡xtan⁡xdxPut sec⁡x=usec⁡x⋅tan⁡x⋅dx=du=∫un−1du=unn=sec⁡nxn+c\begin{aligned} & \int \sec ^n x \tan x d x \\ & =\int \sec ^{n-1} x \cdot \sec x \tan x d x \\ & \text{Put } \sec x=u \\ & \sec x \cdot \tan x \cdot d x=d u \\ & =\int u^{n-1} d u=\frac{u^n}{n}=\frac{\sec ^n x}{n}+c\end{aligned}
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