# Integrals

## Undefined integrals

As with addition and subtraction, multiplication and division, the inverse operation of the derivation is the anti-derivation or undefined integration.

Given a function g (x), any function f '(x) such that f' (x) = g (x) is called an indefinite or anti-derivative integral of f (x).

Examples:

1. If f (x) = , then is the derivative of f (x). One of the antiderivatives of f '(x) = g (x) = x4 é .
2. If f (x) = x3then f '(x) = 3x2 = g (x). One of the undefined antiderivatives or integrals of g (x) = 3x2 is f (x) = x3.
3. If f (x) = x3 + 4, then f '(x) = 3x2 = g (x). One of the undefined antiderivatives or integrals of g (x) = 3x2 is f (x) = x3 + 4.

In examples 2 and 3 we can see that both x3 When x3+4 are undefined integrals for 3x2. The difference between any of these functions (called primitive functions) is always a constant, ie the undefined integral of 3x2 é x3+ C, Where Ç It is a real constant.

### Properties of undefined integrals

The following properties are immediate:

1ª. , that is, the sum or difference integral is the sum or difference of the integrals.

2ª. , ie the multiplicative constant can be taken from the integrand.

3ª. , that is, the derivative of the integral of a function is the function itself.

### Integration by substitution

Be expression .

By substituting u = f (x) for u '= f' (x) or , or, du = f '(x) dx, comes:

,