   Chapter 5.2, Problem 65E ### 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 properties of integrals, together with Exercises 27 and 28, to prove the inequality. ∫ 1 3 x 4 + 1 d x ≥ 26 3

To determine

To prove: The inequality of the integral function 13x4+1dx using the properties of integrals.

Explanation

Given information:

The integral function is 13x4+1dx.

Get the integral function from Exercise 28 in the textbook.

abx2dx=(b3a33) (1)

Calculation:

Check the condition to apply property 7.

Consider the function f(x)=x4+1 and g(x)=x2 with limits  [1,3].

Apply the limits limit [1,3] for the function as given below:

x4+1x4

Take Square root on both sides of Equation (1):

x4+1x4x4+1x2 (2)

Substitute f(x) for x4+1 and g(x) for x2 in Equation (2).

Thus, f(x)g(x) within limits [1,3].

Apply property 7 of integrals:

If f(x)g(x) for axb, then abf(x)dxabg(x)dx

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