   Chapter 7.8, Problem 11E

Chapter
Section
Textbook Problem

# Determine whether each integral is convergent or divergent. Evaluate those that are convergent. ∫ 0 ∞ x 2 1 + x 3   d x

To determine

whether the given integral is convergent or divergent, evaluate it if convergent.

Explanation

Given:

0x21+x3dx.

Formulae used:

d(xn)dx=nxn1xndx=xn+1n+1+c

0x21+x3dx

This is an improper integral so, we take the limit of infinite as t and limit t.

limt0tx21+x3dx (I)

Now assume 1+x3=u

Differentiate with respect to x.

3x2=dudx

So,

x2dx=du3

Now from equation (I) and (II).

limt0tdu3u=limt0tu123du (III)

Taking the integral part of equation (III)

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