To solve the integral $\int 5x^2 \, dx$:

1. Recall the power rule for integration:
   $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \text{ where } n \neq -1.$

2. Apply the rule to $5x^2$:
   $\int 5x^2 \, dx = 5 \cdot \int x^2 \, dx = 5 \cdot \frac{x^{2+1}}{2+1} + C.$

3. Simplify:
   $5 \cdot \frac{x^3}{3} + C = \frac{5}{3}x^3 + C.$

So, the answer is A. $\frac{5}{3}x^3 + C$.

Would you like more details on the steps or related concepts? Here are 5 related questions to consider:

1. How does the power rule work for different values of $n$ in integration?
2. What are the differences between definite and indefinite integrals?
3. Why is the constant $C$ always added in indefinite integrals?
4. How would you integrate a more complex polynomial, such as $x^3 + 2x^2 + x$?
5. How does integration relate to finding areas under curves?

Tip: Always simplify the coefficient and exponent expressions carefully to avoid errors during integration.