Definition of Indefinite Integral

Recovering Position from Velocity

Differentiation turns a position function $s(t)$ into its velocity $v(t)=s'(t)$. Antidifferentiation reverses this process. From $v(t)$, we find every position function whose derivative equals $v(t)$. An initial position is still needed to select one particular position function.

Role of the Constant of Integration

Compare these functions and their derivatives.

All four $F(x)$ functions have the same derivative, $f(x) = 4x$. The reason is that the derivative of a constant, such as $1$, $5$, or $-10$, is zero.

A derivative does not retain the constant from the original function. It cannot tell us whether the original constant was $1$, $5$, $-1000$, or another value. The symbol $C$ represents every possible constant. An indefinite integral therefore includes all antiderivatives that differ only in their constant.

Formal Notation of Integrals

The following notation shows the integrand, the variable of integration, the antiderivative, and the constant of integration:

Each part has a specific role:

• $\int$ is the integral symbol. Its shape comes from an elongated letter S for the Latin word summa, meaning sum.
• $f(x)$ is the function we are going to integrate, called the integrand.
• $dx$ is called the differential of $x$. It shows that the integration is with respect to $x$. Without it, the integral expression is incomplete.
• $F(x) + C$ is the indefinite integral of $f(x)$. This is our answer, where $F(x)$ is the antiderivative and $C$ is the constant of integration.

The defining relationship is:

If the derivative of $F(x)$ is $f(x)$, then the indefinite integral of $f(x)$ includes every function of the form $F(x) + C$.