   Chapter 7.8, Problem 55E

Chapter
Section
Textbook Problem

# The integral ∫ 0 ∞ 1 x ( 1 + x )   d x is improper for two reasons: The interval [ 0 , ∞ ) is infinite and the integrand has an infinite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: ∫ 0 ∞ 1 x ( 1 + x )   d x = ∫ 0 1 1 x ( 1 + x )   d x + ∫ 1 ∞ 1 x ( 1 + x )   d x

To determine

To evaluate:

The given integral by expressing it as a sum of two improper integral of Type 2 and Type 1 (the sum is given).

Explanation

Given:

01x(x+1)dx

Formulae used:

The integration by substitution.

Consider 01x(x+1)dx

Hence, use the formulae of integration by substitution

Now according to the given expression, the value of the given expression 01x(x+1)dx is

01x(x+1)dx=lima0+a12tt(1+t2)dt+limb1b2tt(1+t2)dt

where

t2=x2tdt=dx

01x(x+1)dx=lima0a12tt(1+t2)dt+limb1b2tt(1+t2)dt=2lima0a11(1+t2)dt+2limb1b

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