   Chapter 8.1, Problem 78E

Chapter
Section
Textbook Problem

# Deriving a RuleShow that sec x = sin x cos x + cos x 1 + sin x Then use this identity to derive the basic integration rule ∫ sec x   d x = ln | sec x + tan x | + C .

To determine

To prove: The formula secx=sinxcosx+cosx1+sinx and then apply the specified formulas to derive secxdx=ln|secx+tanx|+C.

Explanation

Given:

The formulas:

secx=sinxcosx+cosx1+sinx

secxdx=ln|secx+tanx|+C

Formula used:

The trigonometric identities:

secx=1cosx

sinxcosx=tanx

sin2x+cos2x=1

The integrations of:

sinxcosxdx=ln|cosx|+C

cosx1+sinxdx=ln|1+sinx|+C

The property of logarithm

ln|a|ln|b|=ln|ab|

Proof:

Consider the term secx

Now, multiply both the numerator and the denominator of secx by 1+sinx,

secx=secx(1+sinx)1+sinx

secx=1+sinxcosx(1+sinx)

Apply the identity sin2x+cos2x=1 in equation (I), to get

secx=(sin2x+cos2x)+sinxcosx(1+sinx)=cos2

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 