# What is the antiderivative of 3 x^2?

#### Understand the Problem

The question is asking for the antiderivative of the function 3x^2, which involves finding a function whose derivative is 3x^2.

The antiderivative of $3x^2$ is $F(x) = x^3 + C$.

The antiderivative of $3x^2$ is $F(x) = x^3 + C$.

#### Steps to Solve

1. Identify the function The function we need to find the antiderivative for is $3x^2$.

2. Apply the power rule for integration To find the antiderivative of $x^n$, we use the power rule: $$\int x^n , dx = \frac{x^{n+1}}{n+1} + C$$ In our case, the constant $3$ can be factored out.

3. Integrate the function We can rewrite the integral: $$\int 3x^2 , dx = 3 \int x^2 , dx$$ Now apply the power rule: $$= 3 \cdot \frac{x^{2+1}}{2+1} + C$$ $$= 3 \cdot \frac{x^3}{3} + C$$

4. Simplify the result Now simplify the expression: $$= x^3 + C$$

5. State the final antiderivative So, the antiderivative of $3x^2$ is: $$F(x) = x^3 + C$$

The antiderivative of $3x^2$ is $F(x) = x^3 + C$.

When finding antiderivatives, it's essential to remember the constant of integration $C$. This constant represents any horizontal shift in the family of functions because the derivative of a constant is zero. As a result, multiple functions can have the same derivative if they differ by a constant.
• Forgetting to include the constant of integration $C$ after finding the antiderivative.