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.