   Chapter 5, Problem 56RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use the properties of integrals to verify the inequality. ∫ 0 1 x sin − 1 x   d x ≤ π / 4

To determine

To verify: The inequality of the integral function using the properties of integrals.

Explanation

Given information:

The integral function is 01xsin1x.

The region lies between x=0 and x=1.

Calculation:

Show the property 4 of the integral as follows:

abcf(x)dx=cabf(x)dx

Where, c is any constant.

Show the integral function as follows:

01xsin1x (1)

Show the method of integration by parts as shown below:

udv=uvvdu (2)

Consider u=sin1x (3)

Differentiate both sides of the Equation (3).

du=11x2dx

Consider dv=xdx

Integrate both sides of the Equation.

v=x22

Apply method of integration by parts in Equation (1) as shown below.

01xsin1xdx=[sin1x×x22]0101x22×11x2dx=[sin1(1)×1220]01x221x2dx=12×π21201x21x2dx=π41201x21x2dx (4)

Modify Equation (3) as shown below.

01xsin1xdx=π41201x21+11x2dx=π41201(11x2(1x2)1x2)dx=π41201(11x21x2)dx=π41201(11x2)dx+1201(1x2)dx (5)

Modify Equation (3)

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