Chapter 8.8, Problem 47E

### 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. ∫ 0 ∞ 4 x  ( x + 6 )   d x

To determine
Whether the improper integral 04x(x+6)dx converges or diverges.

Explanation

Consider the provided integral,

âˆ«0âˆž4x(x+6)dx=âˆ«014x(x+6)dx+âˆ«1âˆž4x(x+6)dx

Now, use the formula:

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

Then,

âˆ«0âˆž4x(x+6)dx=limbâ†’0+âˆ«b14x(x+6)dx+limcâ†’âˆžâˆ«1c4x(x+6)dx

Now, to solve the integral âˆ«4x(x+6)dx apply the formula:

âˆ«1x2+a2=1aarctan(xa)

Let x=u. Then, 12xdx=du

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