Chapter 6.4, Problem 16E

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

10th Edition
Ron Larson
ISBN: 9781305860919

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 7-20, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. SeeExamples 1, 2, and 3. ∫ − ∞ 0 x x 2 + 1 d x

To determine

Whether the improper integral 0xx2+1dx diverges or converges and evaluate if it converges.

Explanation

Given Information:

The provide expression is, 0xx2+1dx.

From definition of improper integral.

bf(x)dx=limaabf(x)dx

Also, the expression for the integration of a inverse function is as follows:

1xdx=|x|+C

The improper integral converges if the limit exists otherwise the improper integral diverges.

Consider the provided expression:

0xx2+1dx

Use the property of improper integral and simplify as:

0xx2+1dx=limbb0xx2+1dx

Integrate the integrand by substation method as:

Assume x2+1=u

Differentiate as:

2xdx=duxdx=12du

Now, substitute the values and integrate by using the inverse function formula as:

xx2+1dx=121udu=12|u|+C

Again, substitute the value of u

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