Recent

Saturday, September 23, 2017

INTRODUCTION OF INDEFINITE INTEGRATION

INTEGRATION

  Definition

 The process of finding a function, given its derivative, is called anti- differentiation
  (or integration). If '(x) = f(x), we say F(x) is an anti-derivative of f(x).

  Examples
 F(x) = -sin x is an anti-derivative of cos x, and ex is an anti-derivative of ex.

Note that if F(x) is an anti-derivative of f(x) then F(x) + c, where c is a constant (called the constant of integration) is also an anti-derivative of f(x), as the derivative of a constant function is 0. In fact they are the only anti-derivatives of f(x).

Integration


We write the indefinite integral off(xdx = F(x) + c.

if F'(x) = f(x) . We call this the indefinite integral of f(x) .

Now you are familiar with the basic concept of Integration.Its time to learn about types of Integration.There are two types of Integration-
(i) Indefinite Integral
(ii) Definite Integral

Indefinite Integral 

what are indefinite integrals? When you learned derivatives you were supposed to solve the following problem. Given the function f(x), find the function F(x) = f'(x). With indefinite integrals we'll solve the reverse problem.
For example, Take a function:
f(x)=2x

with the knowledge of derivative,you can easily find that it is derivative of -
F(x)= x 2
That's because
F'(x)=2x
The function F(x) is called anti-derivative(or primitive or integral) of f(x)
Let's take some other functions.....

F 1 (x)= x 2 +9 F 2 ( x )= x 2 +107
We also have F 1 '(x)=2x F 2 '( x )=2x
So, F1 and F2 are also primitives of f(x). In fact, any function of the form:

F 1 (x)=2 x 2 +C

is a anti-derivative (or primitive or integral) of f(x), where C is a constant.

If we want to find all the integrals of a function, We just need to find one integral and then all other integrals will be equal to that integral plus a constant.

The indefinite integral of a function is just the set of all the primitives of that function.
To denote the indefinite integral of a function f(x) we write:


f( x ) dx
This will be read as " INDEFINITE INTEGRAL OF f(x) with respect to x"

How to find integral?

General formula to evaluate integration -
x n dx= x n+1 n+1 +c
Example
Q1. Find the integral of 

f(x)= x 3 Ans-   x 3 dx= x 3+1 3+1 +c = x 4 4 +c

For all Integration Formulae- Download Integration Formula list

No comments:

Post a Comment