   Chapter 6, Problem 55RE ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Evaluating an Improper Integral In Exercises 51–56, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ∫ 1 ∞ ln   x x   d x

To determine

To calculate: The value of improper integral 1lnxxdx if it converges.

Explanation

Given Information:

The expression is provided as:

1lnxxdx

Formula used:

From definition of improper integral.

If on the interval [a,), the function is continuous, then

af(x)dx=limbabf(x)dx

Here, the improper integral converges if the limit exists and otherwise it diverges.

The integral formula:

1xdx=|x|+C

Calculation:

Consider the provided expression:

1lnxxdx

Use the definition of improper integral af(x)dx=limbabf(x)dx

And simplify as

0e2xdx=limb0be2xdx

Now, in the provided integral,

Assume lnx=u

Now, differentiate

1xdx=du

Now, substitute the values and integrate as:

lnxxdx=udu=u22+C

Again, substitute the value of u

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