The function $\orange {F}$ is an antiderivative of the function $\blue f$ if

$$\orange F'(x)=\blue f(x)$$

We denote the antiderivative of $\blue f$ as follows:

$$\begin{array}{rcl}\displaystyle \int \blue {f(x)} \; \dd x \end{array}$$

This is also called the indefinite integral.

The result of the indefinite integral are functions of the form $\orange F(x) + \green C$ where $\orange F$ is an antiderivative of $\blue f$ and $\green C$ is a constant, because the constant is eliminated when differentiating.

We call $\green C$ the constant of integration.

$$\begin{array}{rcl}\blue f(x)&=&3x^2 \\ \text{gives i.e.} \\ \orange F(x)&=&x^3 \\ \orange F(x)&=&x^3 + \green{3} \\ \orange F(x) &=&x^3 + \green{5} \\ \\ \text{hence} \\\displaystyle \int \blue {3x^2} \; \dd x &=& x^3+\green{C} \\ \text{because} \\ \dfrac{\dd}{\dd x} (x^3+\green C) &=& 3x^2\end{array}$$