Integration of  a function is the inverse(or reverse) process of differentiation. If the derivative f1(x)  of a function f(x) is given, then the process of finding the fuction f(x) is called integration.

Integral as Anti-derivative. We know that `d(x^2)/dx` = 2x`d/dx`(sinx) = cosx, that is 2x is the derivative of x2 and cosx is the derivative of sinx. In this case, we say that x2 is an integral of 2x and sinx is an in integral of cosx.

 

Definition: In general, if {F(x)} = f(x), for all x in a certain interval, then we say that F(x) is a primitive or an integral of f(x) that is F(x) is the integral of f(x) if and only if f(x) is the derivative of F(x). This is why integral ia called anti-derivative.

If F(x) is the integral of f(x), symbolic we write `int` f(x) dx = F(x)..

Here the symbol `int` is represented for integral.

 

Some results of an Anti-derivative

1) `int`ex   dx = ex

2) `int` cosx dx = sinx

3)  int  ( 1 / (1+ x^2 ))  dx =   Tan^-1 x

4) int 1/2sqrt(x) dx = sqrt(x)

But a fuction can have more than one integral as shown below:

 d(e^x)/dx = e^x; d(e^x + 1)/dx = e^x ; d(e^x + c)/dx = e^x; where c is the constant.

 there fore int e^x dx = e^x, int e^x dx = e^x + 1, int e^x dx = e^x + c.

Thus, e^x, e^x + 1, e^x +c are all integral of e^x. In this case, we can write int e^x dx = e^x +c, where c is a constant.

In general, if int f(x) dx = F(x), then we can also write

int f(x) dx = F(x) + c, where c is a constant.

 

 

Some examples of an Anti-derivative

Ex:1  int x2 dx = `x^(2+1)/(2+1)` + c

Sol:

= `1/3`

Ex:2: int sqrt(x) = int x^(1/2) dx

Sol:

= (x^ (1/2 +1))/(1/2 + 1) +c = 2/3 x^(3/2) +c.

Practice problems on Anti - derviative:

Q:1 int 1/ x^3 dx

Ans: - 1/2x^2 +c

Q:2: int( 3x^2 - (1/2)sqrt(x)) dx

Ans: x^3 - (1/3) x^(3/2) + c