   Chapter 8.8, Problem 47E

Chapter
Section
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,

04x(x+6)dx=014x(x+6)dx+14x(x+6)dx

Now, use the formula:

abf(x)dx=limcbacf(x)dx+limca+cbf(x)dx

Then,

04x(x+6)dx=limb0+b14x(x+6)dx+limc1c4x(x+6)dx

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

1x2+a2=1aarctan(xa)

Let x=u. Then, 12xdx=du

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