Problem 26 - Essential Calculus

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{\infty} \frac{e^{x}}{1+e^{x}} d x $$

Step-by-Step Solution

Answer

The improper integral \(\int_{0}^{\infty} \frac{e^{x}}{1+e^{x}} dx \) diverges. It does not evaluate to a finite value.

Step 1: Define the Function and the Limit for the Improper Integral

Start by defining the function for which the integral is sought: \(f(x) = \frac{e^{x}}{1+e^{x}}\). Then, define the limits for the improper integral: \(\lim_{{b \to \infty}} \int_{{0}}^{{b}} f(x) dx \). This transformation positions the problem to be solved through limiting processes. Setting it up like this creates the problem: \(\lim_{{b \to \infty}} \int_{{0}}^{{b}} \frac{e^{x}}{1+e^{x}} dx \).

Step 2: Apply Substitution

For the function \( f(x) = \frac{e^{x}}{1+e^{x}}\), let \(u = 1 + e^x\). Then, differentiating gives \( du = e^x dx \). Rewrite the integral in terms of \(u\): \(\lim_{b \to \infty} \int_{1}^{1+e^{b}} \frac{1}{u} du \).

Step 3: Apply Formula for Definite Integral

The definite integral of \(1/u\) from 1 to \(1+e^{b}\) is the natural logarithm of the upper limit divided by the lower limit. Substituting \(1+e^{b}\) back in for \(u\), we get \(\lim_{b \to \infty} [\ln(1+e^{b}) - \ln(1)]\)

Step 4: Simplify and Evaluate the Limit

Simplify the limit to get \(\lim_{b \to \infty} \ln(1+e^{b})\). As \(b \to \infty\), \(1+e^{b}\) also tends to infinity, and thus \(\ln(1+e^{b})\) tends to infinity as well. So the limit, and thus the integral, does not exist or diverges.