The Definition of Antiderivative

Antiderivative

In the former sections, we discussed how to approximate the rate of the change of a given function. From the other side of the problem, we sometimes want to know that how to find the origin function using its rate of change. That comes to the definition of the antiderivative.

Take ${\displaystyle f(x)=2x}$ as an example, suppose that there is a ${\displaystyle F(x)}$ satisfies that ${\displaystyle F^{\prime }(x)=f(x)}$, then, from the power rule of the derivative, we can know that ${\displaystyle F(x)=x^2}$ is a reasonable answer.

Since the process is the inverse of the derivative, we call it antiderivative.

Non-Uniqueness

But is ${\displaystyle F(x)=x^2}$ the only antiderivative of ${\displaystyle f(x)}$?

The answer is no. We can know that ${\displaystyle f(x)}$ is the derivative of ${\displaystyle F(x)}$ which only shows the rate of change of ${\displaystyle F(x)}$, but not the exact value at the exact point. While the derivative of a constant is always ${\displaystyle 0}$, ${\displaystyle F(x)}$ with any constant ${\displaystyle C}$ can be the origin function of ${\displaystyle f(x)}$. We call all ${\displaystyle F(x)+C}$ the family of the origin functions of ${\displaystyle f(x)}$.

Integral

The family of the origin functions is the indefinite integral of the function, denoted by ${\displaystyle \int }$. That is:

${\displaystyle \int f(x)dx=F(x)+C}$

Roughly, the change of the indefinite integral in the given interval is the definite integral, since the constant ${\displaystyle C}$ is eliminated in the subtraction.