Indefinite Integrals : In the past two chapters we've been given a function, f ( x ) , and asking what the derivative of this function was.  Beginning with this section we are now going to turn things around.  Now we desire to ask what function we differentiated to get the function f ( x ) .

Definitions (anti-derivative, integral symbol, integrand, integration variable)

A function, f ( x ) , an anti-derivative of f ( x ) is any function  F ( x ) such that

                                                       F ′ ( x ) = f ( x )

If F ( x ) is a anti-derivative of f ( x ) then the most general anti-derivative of f ( x ) is called an indefinite integral and specified,

              ∫ f ( x ) dx = F ( x ) + c, c is any constant

In this definition the ∫ is called as the integral symbol, f (x) is called the integrand, x is called as the integration variable and the "c" is called the constant of integration.

                Note as well that frequently we will just say integral instead of indefinite integral (or definite integral for which matter while we get to those).  It will be apparent from the context of the problem that we are talking regarding an indefinite integral (or definite integral).

The procedure of finding the indefinite integral is known as integration or integrating f(x).  If we have to be specific regarding the integration variable we will say that we are integrating f(x) w.r.t. x.

Example   Evaluate the indefinite integral.

∫ x⁴ + 3x - 9 dx

Solution

As it is really asking for the most general anti-derivative we just require reusing the final answer from the first example.

The indefinite integral is,

∫ x⁴ + 3x - 9 dx= 1/5 x⁵ + (3/2) x² - 9x + c