To integrate the function $\frac{5x}{1 + 3x^2}$, we can perform a substitution to simplify the integral. Here's the step-by-step process:

Step 1: Substitution

Let: $u = 1 + 3x^2$ Then, differentiate $u$ with respect to $x$: $du = 6x \, dx$ This implies: $\frac{du}{6} = x \, dx$

Step 2: Rewriting the Integral

Now, substitute into the integral: $\int \frac{5x}{1 + 3x^2} \, dx = \int \frac{5}{u} \cdot \frac{du}{6}$ Simplify the constant factor: $= \frac{5}{6} \int \frac{1}{u} \, du$

Step 3: Integration

We know that: $\int \frac{1}{u} \, du = \ln |u|$ So the integral becomes: $\frac{5}{6} \ln |u| + C$

Step 4: Back-Substitute $u$

Substitute $u = 1 + 3x^2$ back into the result: $= \frac{5}{6} \ln |1 + 3x^2| + C$

Thus, the final answer is: $\int \frac{5x}{1 + 3x^2} \, dx = \frac{5}{6} \ln |1 + 3x^2| + C$

Would you like more details on this process or have any other questions? Here are some related questions to expand on this topic:

1. How do you handle more complex rational functions in integration?
2. Can you explain the process of using trigonometric substitution in integrals?
3. What are other methods of integration, like partial fractions, and when are they used?
4. How do you apply integration by parts to functions like this?
5. Could you explain the concept of the natural logarithm function in the context of integration?

Tip: Remember that substitution is a powerful method for simplifying integrals that involve functions like $1 + 3x^2$, which can lead to a simpler integral.