First slide

Methods of integration

Question

$If J = \int \sec^{3} x d x then J is equal to$

Moderate

A

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

$\sec x \tan x - \log |\sec x + \tan x| + C$
C

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

$\frac{1}{2} [s e c x \tan x + \log |\sec x + \tan x|] + C$

Solution

$Let J = \int \sec^{3} x d x$
$J = \int \sec x \sec^{2} x d x$
Using by parts
$\begin{array}{l} J = \sec x \int \sec^{2} d x - \int [\frac{d}{d x} (\sec x) \int \sec^{2} x d x] d x \\ J = \sec x \tan x - \int \sec x \tan x \tan x d x \\ J = \sec x \tan x - \int \sec x (\sec^{2} x - 1) d x \\ 2 J = \sec x \tan x + \log | \sec x + \tan x | \\ J = \frac{1}{2} [\sec x \tan x + \log | \sec x + \tan x |] \end{array}$

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