First slide

Introduction to integration

Question

$\int {1 + 2 \tan x (\tan x + \sec x)}^{1 / 2} d x =$

Easy

A

$\log (\sec x + \tan x) + c$
B

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

$\log \sec x (\sec x + \tan x) + c$
D

$None of these$

Solution

$\begin{array}{l} \int {(1 + 2 \tan^{2} x + 2 \tan xsec x)}^{1 / 2} dx \\ = \int {(\sec^{2} x + \tan^{2} x + 2 \tan xsec x)}^{1 / 2} dx \\ = \int (\sec x + \tan x) dx \\ = \log (\sec x + \tan x) + \log \sec x + c \\ = \log \sec x (\sec x + \tan x) + c \end{array}$

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