# BHARDWAJ CLASSES

## 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).

We write f(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-   $∫ x 3 dx = x 3+1 3+1 + c = x 4 4 + c$