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To solve this problem, we need to evaluate the integral:

\[
\int 7x^2 \sqrt{12 + 4x^3} \, dx
\]

The best approach here is to use substitution.

### Step 1: Set up the substitution

Let:
\[
u = 12 + 4x^3
\]

Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = 12x^2
\]

Thus:
\[
dx = \frac{du}{12x^2}
\]

### Step 2: Substitute into the integral

Now, rewrite \( 7x^2 \sqrt{12 + 4x^3} \, dx \) in terms of \( u \):

\[
7x^2 \sqrt{u} \cdot \frac{du}{12x^2}
\]

The \( x^2 \) terms cancel, leaving:

\[
\frac{7}{12} \int \sqrt{u} \, du
\]

### Step 3: Integrate with respect to \( u \)

Rewrite \( \sqrt{u} \) as \( u^{1/2} \) and integrate:

\[
\frac{7}{12} \int u^{1/2} \, du = \frac{7}{12} \cdot \frac{u^{3/2}}{3/2} + C
\]

Simplifying, we get:

\[
\frac{7}{12} \cdot \frac{2}{3} u^{3/2} + C = \frac{7 \cdot 2}{12 \cdot 3} u^{3/2} + C = \frac{14}{36} u^{3/2} + C = \frac{7}{15} u^{3/2} + C
\]

### Step 4: Substitute back \( u = 12 + 4x^3 \)

Replace \( u \) with \( 12 + 4x^3 \):

\[
\frac{7}{15} (12 + 4x^3)^{3/2} + C
\]

### Solution

The correct answer is:

\[
\frac{7}{15} (12 + 4x^3)^{3/2} + C
\]

Thus, the answer is **A**.

---

Do you want a deeper explanation of any step, or have any questions?

### Related Questions

1. How would you approach this integral if it were \( \int x^2 \sqrt{12 + 4x^3} \, dx \) instead?
2. Can substitution be used for all integrals with square roots?
3. How does the chain rule relate to integration by substitution?
4. What if the integral were \( \int x^2 (12 + 4x^3)^{5/4} \, dx \)?
5. How would integration change if we used a different constant in the substitution?

### Tip
For integrals involving square roots, try substitution where the square root's argument can be simplified. This often turns the integral into a power rule form.

Evaluate the integral: ∫ 7x^2 √(12 + 4x^3) dx

Learn how to evaluate the integral ∫ 7x^2 √(12 + 4x^3) dx using substitution. This solution includes key calculus concepts, step-by-step explanations, and demonstrates the application of the substitution method in integration. Suitable for college-level calculus students.

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