Chapter 7.1, Problem 53E

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use integration by parts to prove the reduction formula.53. ∫ tan n x   d x   =   tan n − 1 x n   −   1   −   ∫ tan n − 2 x   d x   ( n ≠ 1 )

To determine

To prove: the reduction formula by using integration by parts.

Explanation

Given information:

The integral function is âˆ«tannxdx=tannâˆ’1xnâˆ’1âˆ’âˆ«tannâˆ’2xdx_.

Calculation:

Show the integral function as shown below:

âˆ«tannxdx (1)

Since n is not equal to 1.

Modify Equation (1).

âˆ«tannxdx=âˆ«tannâˆ’2xtan2xâ€‰dx=âˆ«tannâˆ’2x(sec2xâˆ’1)â€‰dx=âˆ«tannâˆ’2xsec2xâ€‰dxâˆ’âˆ«tannâˆ’2â€‰xdx (2)

Show the method of integration by parts as shown in below:

âˆ«udv=uvâˆ’âˆ«vdu (3)

Consider the function u=tannâˆ’2x (4)

Differentiate both sides of the Equation (4).

du=(nâˆ’2)tannâˆ’3xsec2xdx

Consider dv=sec2xdx

Integrate both sides of the Equation.

v=tanx

Apply method of integration by parts as shown below.

Substitute tannâˆ’2x for u, sec2xdx for dv, tanx for v, and (nâˆ’2)tannâˆ’3xsec2xdx for du in Equation (2)

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

Use the guidelines of Section 4.5 to sketch the curve. y=e2xx2

Single Variable Calculus: Early Transcendentals, Volume I

Expand each expression in Exercises 122. (2x3)2

Finite Mathematics and Applied Calculus (MindTap Course List)

In Problems 15-18, find the second partials.

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate .

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th