To solve the integral

$\int_{0}^{1} \frac{3x^2}{4x^3 + 1} \, dx,$

we can use a substitution method.

Step 1: Substitution

Let $u = 4x^3 + 1$. Then, differentiate $u$ with respect to $x$:

$\frac{du}{dx} = 12x^2 \quad \text{or} \quad du = 12x^2 \, dx.$

Notice that $3x^2 \, dx = \frac{du}{4}$.

Step 2: Change the limits of integration

When $x = 0$, $u = 4(0)^3 + 1 = 1$.

When $x = 1$, $u = 4(1)^3 + 1 = 5$.

Step 3: Substitute and simplify

Substitute $u$ into the integral:

$\int_{0}^{1} \frac{3x^2}{4x^3 + 1} \, dx = \int_{1}^{5} \frac{1}{4u} \, du.$

Step 4: Integrate

Now, integrate with respect to $u$:

$\int_{1}^{5} \frac{1}{4u} \, du = \frac{1}{4} \int_{1}^{5} \frac{1}{u} \, du = \frac{1}{4} \left[ \ln |u| \right]_{1}^{5}.$

Step 5: Evaluate the definite integral

Evaluate the expression:

$\frac{1}{4} \left[ \ln(5) - \ln(1) \right] = \frac{1}{4} \ln(5).$

Thus, the final answer is:

$\boxed{\frac{1}{4} \ln(5)}.$

Would you like any more details or have any questions?

Here are some related questions to explore:

1. How would the integral change if the limits were from 0 to 2 instead of 0 to 1?
2. Can this problem be solved using any other integration technique, such as integration by parts?
3. What happens to the integral if the numerator was $6x$ instead of $3x^2$?
4. How does the substitution method change if the integral had a different denominator, such as $5x^3 + 2$?
5. What are some common integrals that are solved using substitution?

Tip: Substitution is a powerful technique for solving integrals where the integrand contains a function and its derivative.