First slide

Introduction to integration

Question

$Evaluate \int \frac{\sec x}{\sec x + \tan x} d x .$

Easy

A

$= \tan x - \sec x + c$
B

$= \tan x + \sec x + c$
C

$= - \tan x - \sec x + c$
D

$= \sec x - \tan x + c$

Solution

$\begin{array}{l} \int \frac{\sec x}{\sec x + \tan x} d x \\ = \int \frac{\sec x (\sec x - \tan x)}{(\sec x + \tan x) (\sec x - \tan x)} d x \\ = \int \frac{\sec^{2} x - \sec x \tan x}{\sec^{2} x - \tan^{2} x} d x \\ = \int \sec^{2} x d x - \int \sec x \tan x d x \\ = \tan x - \sec x + c \end{array}$

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