Chapter 8.3, Problem 25E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Finding an Indefinite Integral Involving Secant and Tangent In Exercises 21–34, find theindefinite integral. ∫ tan 5 x 2 d x

To determine

To calculate: The value of the following indefinite integral tan5x2dx

Explanation

Given: The provided indefinite integral is âˆ«tan5x2dx

Formula Used:

The following trigonometric identity,

tan2x2=sec2x2âˆ’1

The following concepts based on differentiation,

ddxtanx2=12sec2x2.

ddxcosx2=âˆ’12sinx2

Calculation:

Consider the provided integral expression âˆ«tan5x2dx.

âˆ«tan5x2dx=âˆ«tan2x2tan3x2dx=âˆ«(sec2x2âˆ’1)tan3x2dx=âˆ«(sec2x2tan3x2âˆ’tan3x2)dx=âˆ«(sec2x2tan3x2)dxâˆ’âˆ«(tan2x2tanx2)dx

Simplify further,

âˆ«tan5x2dx=âˆ«(sec2x2tan3x2)dxâˆ’âˆ«(sec2x2âˆ’1)tanx2dx=âˆ«(sec2x2tan3x2)dxâˆ’âˆ«(sec2x2tanx2âˆ’tanx2)dx âˆ«tan5x2dx=âˆ«(sec2x2tan3x2)dxâˆ’âˆ«(sec2x2tanx2)dx+âˆ«(tanx2)dx ------(1)

Now, integrate this equation by the method of substitution,

Let consider tanx2=y ------(2)

Now differentiate equation (2) with respect to x, to get

ddxtanx2=dydx

### 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