   Chapter 7.R, Problem 79E

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Textbook Problem
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# Use the substitution u = 1 / x to show that ∫ 0 ∞ ln  x 1 + x 2 d x = 0

To determine

To show:

That 0lnx1+x2dx=0 using substitution of u = 1/x.

Explanation

Given:

0lnx1+x2dx

Formulae used:

The integration by using substitution

Consider 0lnx1+x2dx

Hence, use the work done formula F.ds

Now, the value of the integration of the given expression 0lnx1+x2dx is

Let

u=1xdu=1x2dx

Thus, the expression becomes,

0lnx1+x2dx=0ln(1u)1+(1u)2(1u2du)=0u2(ln1lnu)1+(u)2(1u2du)=0lnu1+(u)2</

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