The-Indefinite-Integral-of-a-Function-f-Defined-in-a-Space-D

Indefinite integral of a
function

Stella Seremetaki

Stella Seremetaki Pure Mathematician

Introduction

An indefinite integral of a function f
defined in a subset D of the set of real
numbers is the set of functions where their
derivative equals f(x), for every x in D

If the indefinite integral of f, for every x in
D is g(x)+c, means that:

 f( x)dx  g( x)  c

Example

Applying the definition of the indefinite

integral calculate the integral below for

every real number x

A  x dx
1  x2

Rules of derivation

Let x be a positive real variable

 x  1
2
x

Let g be a function, defined as below

g : R  0,

Suppose that the derivative of g exists for
every real number x

 g( x)   2 1 .g( x)
g( x)

For instance

For every real number x

 1  x2   2 1 (1  x 2 )
  1  x2

 1  x2   x
  1  x2

Due to the rules of derivation for every real
number x :

  x dx  1  x2  c
1  x2

Thank you!