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answer this question with as minimal steps as possible 

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.

answer this question with as minimal steps as possible

Learn how to solve the integral of (3x^2) / (4x^3 + 1) from 0 to 1 using the substitution method. This problem covers key concepts in integration, including the substitution rule, and is suitable for students in Grades 11-12.

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