First slide

Methods of integration

Question

$I = \int \sec^{3} x d x is equal to [x \in (0, \frac{π}{2})]$

Moderate

A

$\frac{\tan x \sec x}{2} + \frac{1}{2} \log |\sec x - \tan x| + C$
B

$\frac{\tan x \sec x}{2} + \frac{1}{2} \log |\sec x + \tan x| + C$
C

$\frac{\tan x \sec x}{2} + \frac{1}{2} \log |\sqrt{\sec x + \tan x}| + C$
D

$\frac{\tan x \sec x}{2} - \frac{1}{2} \log |\sqrt{\sec x + \tan x}| + C$

Solution

$Let I = \int \sec^{3} x d x = \int \sec x \sec^{2} x d x$
$I = \int \sqrt{1 + \tan^{2} x} \sec^{2} x d x$
$Put \tan x = t \Rightarrow \sec^{2} x d x = d t$
$\begin{array}{l} = \int \sqrt{1 + t^{2}} d t = [\frac{t}{2} \sqrt{1 + t^{2}} + \frac{1}{2} \log |t + \sqrt{t^{2} + 1}|] + C \\ = \frac{\tan x}{2} \sqrt{1 + \tan^{2} x} + \frac{1}{2} \log |\tan x + \sqrt{\tan^{2} x + 1}| + C \\ = \frac{\tan x \sec x}{2} + \frac{1}{2} \log | \tan x + \sec x | + C \end{array}$

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