Chapter 6, Problem 56RE

### 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
1 views

# Evaluating an Improper Integral In Exercises 51–56, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ∫ 0 ∞ e x 1 + e x   d x

To determine

To calculate: The value of improper integral 0ex1+exdx if it converges.

Explanation

Given Information:

The provided expression is,

0ex1+exdx

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:

eaxdx=eaxa+C

Here, a0

Calculation:

Consider the provided expression:

0ex1+exdx

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

And simplify as

0ex1+exdx=limb0bex1+exdx

In the provided integral

Assume 1+ex=u

Differentiate

exdx=du

Now, substitute the values and integrate by the us

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