### 3.1.1. Antiderivative Function and Indefinite Integral

In differential calculus, given a function $F(x)$, one finds its derivative $f(x)=F^{\prime}(x)$. In practice, it is often necessary to solve the inverse problem: to restore the function $F(x)$, knowing its derivative $f(x)$. In this case, the function $F(x)$ is called the antiderivative of $f(x)$.

Definition 1. A function $F(x)$ is called an antiderivative of the function $f(x)$ if the derivative of $F(x)$ is equal to $f(x)$, i.e.,

$$
F^{\prime}(x)=f(x) \text { or } d F(x)=f(x) d x .
$$

Example. Suppose we have the function $f(x)=2 x$. The function $F(x)=x^{2}$ is an antiderivative of $f(x)$, because $\left(x^{2}\right)^{\prime}=2 x$. But the function $F(x)=x^{2}+1$ is also an antiderivative of $f(x)$, since $\left(x^{2}+1\right)^{\prime}=2 x$, and generally - any function $F(x)=x^{2}+C$ (where $C$ is an arbitrary constant) is an antiderivative of $f(x)$. Thus, this function has a set of antiderivatives, and it can be shown that any two of them differ by a constant number.

Theorem 1 (on two antiderivatives). If $F_{1}(x)$ and $F_{2}(x)$ are two antiderivatives of the function $f(x)$ on the interval $[a, b]$, then the difference between them is a constant number.

From this theorem, it follows that if any antiderivative $F(x)$ of the function $f(x)$ is found on some interval (finite or infinite), then any other antiderivative of this function can be found using the formula

$$
F(x)=F(x)+C,
$$

where $C=$ const.

Definition 2. The set of antiderivatives of the function $f(x)$ is called the indefinite integral of the function $f(x)$ and is denoted by the symbol $\int f(x) d x$. Thus, by definition, we have:

$$
\int f(x) d x=F(x)+C
$$

where $F(x)$ is some antiderivative; $C=$ const; $x$ is the independent variable of integration; $f(x)$ is the integrand; $f(x) d x$ is the integrand expression; $\int$ is the integral sign.

The operation of finding antiderivatives of a function is called its integration.

Geometric meaning of the indefinite integral. The graph of the antiderivative $F(x)$ is called the integral curve. In the coordinate system $x 0 y$, the graphs of all antiderivatives of a given function represent a family of curves depending on the value of the constant $C$ and obtained from each other by parallel translation along the $0 y$ axis. For the example considered above, we have:

$$
\int 2 x d x=x^{2}+C
$$

The family of antiderivatives ($x^{2}+C$) is geometrically interpreted as a set of parabolas.

If one needs to find one from the family of antiderivatives, additional conditions are set to determine the constant $C$. Usually, this is done by setting initial conditions: at the argument value $x=x_{0}$, the function has the value $F\left(x_{0}\right)=y_{0}$.

Example. It is required to find that antiderivative of the function $y=2 x$ which takes the value 3 at $x_{0}=1$.
Solution. $\int 2 x d x=x^{2}+C ; 1^{2}+C=3 ; C=2$.

The desired antiderivative: $F(x)=x^{2}+2$.

Theorem 2 (on the existence of an indefinite integral).

If the function $f(x)$ is continuous on the interval $[a, b]$, then an indefinite integral of this function exists on the same interval.