Chapter 8.8, Problem 45E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Evaluating an Improper Integral In Exercises 33–48, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. ∫ 3 ∞ 1 x x 2 − 9   d x

To determine
Whether the improper integral 31xx29dx converges or diverges.

Explanation

Consider the provided integral,

âˆ«3âˆž1xx2âˆ’9dx=âˆ«3âˆž1xx2âˆ’32dx

Now, use the formula:

âˆ«abf(x)dx=limcâ†’aâˆ’âˆ«acf(x)dx+limcâ†’a+âˆ«cbf(x)dx

Then,

âˆ«3âˆž1xx2âˆ’32dx=âˆ«351xx2âˆ’32dx+âˆ«5âˆž1xx2âˆ’32dx=limaâ†’3+âˆ«a51xx2âˆ’32dx+limbâ†’âˆžâˆ’âˆ«5b1xx2âˆ’32dx

Now, apply the formula:

âˆ«axx2âˆ’a2dx=arcsec|xa|

Then,

âˆ«3âˆž1xx2âˆ’32dx=lim

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