What is the integral of f(x) = 3x²?

To determine the integral of the function f(x) = 3x², we apply the power rule of integration. This rule states that the integral of x raised to any power n is given by (x^(n+1))/(n+1) plus a constant of integration C.

In this case, the function is 3x². First, we focus on the variable part, which is x². According to the power rule, when we integrate x², we increase the exponent by one (from 2 to 3) and divide by the new exponent:

∫3x² dx = 3 * (x^(2+1))/(2+1) + C

= 3 * (x³/3) + C

= x³ + C.

Thus, the integral of 3x² is indeed F(x) = x³ + C, which makes this option the correct choice.

Understanding the correct integral is rooted in recognizing how integration increases the degree of each term and applies the necessary coefficients, leading to the appropriate expression that includes the constant of integration.