Step 1: Identify the integral. The problem asks to evaluate the indefinite integral of $3x^2$ with respect to $x$. $$ \int 3x^2 dx $$ Step 2: Apply the constant multiple rule for integration. The constant multiple rule states that $\int c \cdot f(x) dx = c \int f(x) dx$. Here, $c=3$ and $f(x)=x^2$. $$ \int 3x^2 dx = 3 \int x^2 dx $$ Step 3: Apply the power rule for integration. The power rule for integration states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$. Here, $n=2$. $$ 3 \int x^2 dx = 3 \left( \frac{x^{2+1}}{2+1} \right) + C $$ $$ = 3 \left( \frac{x^3}{3} \right) + C $$ Step 4: Simplify the expression. $$ = \frac{3x^3}{3} + C $$ $$ = x^3 + C $$ The final answer is $\boxed{x^3 + C}$.