First slide

Methods of integration

Question

$\int \frac{\sec x}{{(\sec x + \tan x)}^{2}} d x =$

Moderate

A

$\frac{- 1}{2 {(\sec x + \tan x)}^{2}} + c$
B

$\frac{1}{{(\sec x + \tan x)}^{2}} + c$
C

$\frac{1}{2 {(\sec x - \tan x)}^{2}} + c$
D

$\frac{1}{2 {(\sec x + \tan x)}^{2}} + c$

Solution

                   $\sec x + \tan x = t$
$\sec x (\sec x + \tan x) d x = d t$

                           $\sec x d x = \frac{d t}{t}$

                            $\int \frac{1}{t^{2}} \frac{d t}{t} = \int \frac{1}{t^{3}} d t$

                                          $= \frac{- 1}{2 t^{2}} + C$

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