Courses in Math - Definition - Indefinite Integral

This page contains definition of indefinite integral. Indefinite integral of a function is a one of the most important mathematical properties. Integral represents a function of which a given function is the derivative. Indefinite integral is an integral expressed without limits. Because of this an indefinite integral contains an arbitrary constant. The integral calculus's fundamentals are based on the determination, properties and application of integrals.

Indefinite integrals. Definition.

A function F(x) is called antiderivative or indefinite integral of a function f(x) in a given range Ω if in the whole range the function f(x) is the derivative of the function F(x)

F'(x) = f(x) in Ω

or equivalently f(x)dx is the differential of F(x)

dF(x) = f(x)dx in Ω

The process of finding all indefinite integrals of a given function is called integration.

Theorem

If in a given range Α (finite or infinite, closed or not) F(x) is an antiderivative of a function f(x) then a function F(x) + C where C is a constant is also an antiderivative of f(x). Oppositely, each antiderivative of f(x) in the range Α can be represented by F(x) + C.

Proof

The first part of the theorem:

[F(x) + C]' = F'(x) = f(x)

the second part of the theorem: Assume that a function Φ(x) is an antiderivative of f(x) i. e.

Φ(x)' = f(x)

Because both functions Φ and F(x) have the same derivative in the range Α they must differ of a constant value C1

Φ(x) = F(x) + C1

The expression F(x) + C where C is a constant is the general form of a function which derivative equals f(x). This expression is called indefinite integral of f(x) and is denoted as

 ∫ f(x)dx

Much more about indefinite integrals you can find at the page indefinite integrals - examples of analytical integration.