The integral you're asking for is:

\[
\int \frac{x}{x^2 + 3} \, dx
\]

To solve this, we can use a substitution method.

### Step-by-Step Solution:

1. **Substitute**: Let \( u = x^2 + 3 \), so \( du = 2x \, dx \).

2. **Rearrange**: Now, divide both sides by 2 to isolate \( x \, dx \):
   \[
   \frac{du}{2} = x \, dx
   \]

3. **Rewrite the integral**: Substituting this into the original integral, we get:
   \[
   \int \frac{x}{x^2 + 3} \, dx = \int \frac{1}{2} \frac{du}{u}
   \]

4. **Solve the integral**: Now, the integral becomes:
   \[
   \frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln |u| + C
   \]

5. **Substitute back**: Since \( u = x^2 + 3 \), we substitute back to get the final result:
   \[
   \frac{1}{2} \ln |x^2 + 3| + C
   \]

Thus, the integral of \( \frac{x}{x^2 + 3} \, dx \) is:

\[
\boxed{\frac{1}{2} \ln |x^2 + 3| + C}
\]

Would you like further details on substitution or solving techniques?

Here are 5 related questions to deepen your understanding:

1. What is the integral of \( \frac{x}{x^2 + 1} \)?
2. How would you integrate \( \frac{1}{x^2 + 3} \)?
3. Can you generalize this method for \( \frac{x}{x^2 + a} \)?
4. How would the integration change if the expression was \( \frac{x^3}{x^2 + 3} \)?
5. How do you evaluate an improper integral involving \( \frac{x}{x^2 + 3} \) over an infinite domain?

**Tip**: Substitution is one of the most powerful techniques in calculus, especially for rational functions. Look for expressions where the derivative of part of the function appears elsewhere in the integral.

Explore how to solve the integral of x/(x^2 + 3) using the substitution method. This step-by-step solution includes key calculus concepts such as substitution, logarithmic integration, and the fundamental theorem of calculus. Ideal for undergraduate calculus students.

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