Integrals, together with Derivatives, are the basic objects of Calculus. Indefinite integrals are defined through Primitives (or Antiderivatives).

A function  F(x)  is called a primitive (or antiderivative) of  f (x) if

for all  x  in the domain of  f (x).

Primitives have the following important property.

Indeed,    by definition.
Therefore, the derivative of the difference between functions  F1(x)  and F2(x)  is equal to zero for all  x  on the given interval:

.

Hence, the difference  F1(x) – F2(x)  equals a constant by the corollary to the Mean Value Theorem.

If a function  f (x)  has one primitive  F(x), then it has an infinite number of primitives. The set of all primitives can be expressed as  F(x) + C, where  C  is an arbitrary constant.

The set of all primitives of  f (x)  is called the indefinite integral of  f (x)  with respect to  x.

if and only if .

The function  f (x)  under the integral sign is called the integrand.

The  x  is the integration variable.

The symbol  dx  is the differential of  x.

An arbitrary constant  C  is said to be a constant of integration.

Note that solutions of differential equations can be expressed in terms of indefinite integral. For example,

.

Indefinite Integrals Definite Integrals Differential Equations