Step 1: Identifying the Function for Integration

We start by identifying the function we aim to integrate, which is \( \frac{1}{2x + 1} \). This function is a rational function with a linear term in the denominator.

Step 2: Setting Up the Integral

We set up the integral with the given limits, which are \( x = 0 \) and \( x = \infty \):

\[ \int_{0}^{\infty} \frac{1}{2x + 1} \, dx \]

Step 3: Applying Limits for the Improper Integral

Since the integral has an infinite upper limit, it is an improper integral. We use limits to evaluate it:

\[ \lim_{{b \to \infty}} \int_{0}^{b} \frac{1}{2x + 1} \, dx \]

Step 4: Finding the Antiderivative

The antiderivative of \( \frac{1}{2x + 1} \) is \( \frac{1}{2} \ln|2x + 1| + C \), where \( C \) is the constant of integration.

Step 5: Applying the Upper and Lower Limits

We apply the upper and lower limits to the antiderivative:

\[ \left[\frac{1}{2} \ln|2x + 1|\right]_{0}^{b} \]

Step 6: Evaluating the Limits

After applying the limits, we find:

\[ \lim_{{b \to \infty}} \left(\frac{1}{2} \ln|2b + 1| – \frac{1}{2} \ln|2 \cdot 0 + 1|\right) = \infty – 0 = \infty \]

Step 7: Final Result and Interpretation

The integral diverges, meaning it does not converge to a finite value. This result is intriguing because it shows that despite the presence of a linear term in the denominator, the function does not have a finite integral over the given range.