## 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:

- If f (x) = , then is the derivative of f (x). One of the antiderivatives of f '(x) = g (x) = x
^{4}é .

- If f (x) = x
^{3}then f '(x) = 3x^{2}= g (x). One of the undefined antiderivatives or integrals of g (x) = 3x^{2}is f (x) = x^{3}.

- If f (x) = x
^{3}+ 4, then f '(x) = 3x^{2}= g (x). One of the undefined antiderivatives or integrals of g (x) = 3x^{2}is f (x) = x^{3}+ 4.

In examples 2 and 3 we can see that both **x ^{3}** When

**x**are undefined integrals for

^{3}+4**3x**. The difference between any of these functions (called

^{2}**primitive functions**) is always a constant, ie the undefined integral of

**3x**é

^{2}**x**, Where

^{3}+ C**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:

,

admitting you know .

The variable substitution method requires the identification of **u** and **u '** or **u** and **du** in the given integral.