   Chapter 7.8, Problem 49E

Chapter
Section
Textbook Problem

# Use the Comparison Theorem to determine whether the integral is convergent or divergent. ∫ 0 ∞ x x 3 + 1   d x

To determine

whether the given integral is convergent or divergent, by using the comparison theorem.

Explanation

Given:

0xx3+1dx

Formulae used:

For the expression 11xpdx, if p>1, then the integral is convergent, and if p1 then the function is divergent.

Theorem used:

Suppose that f and g are continuous function with f(x)g(x)0 for xa.

(a) If af(x)dx is convergent, then ag(x)dx is convergent.

(b) If ag(x)dx is divergent, then af(x)dx is divergent.

Consider 0xx3+1dx

Split the above expression into two integrals from 0 to 1 and 1 to , since to apply the formulae simplification is needed.

Also, direct integration is not easy, since integration from 0 to 1 is finite but not after that.

0xx3+1dx=01xx3+1dx+1xx3+1dx

So,

x3+1x3, it can be written as xx3+1xx3 or xx3+11x2 …… (1)

Now apply the formulae mentioned above and get,

Here, the value of p is 2 and 2>1, therefore p>1

### 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

#### Distance Find the distance between the given numbers. 73.

Precalculus: Mathematics for Calculus (Standalone Book)

#### Find the median for the following set of scores: 1, 9, 3, 6, 4, 3, 11, 10

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

#### In Exercises 107-120, factor each expression completely. 109. 4a2 b2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### If and , sketch the shape of the graph of .

Mathematical Applications for the Management, Life, and Social Sciences

#### Estimate 14(x3x2) using the Left Endpoint Rule and n = 5. a) 203 b) 20 c) 24 d) 30

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