Integration of a function is the inverse(or reverse) process of differentiation. If the derivative f^{1}(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 x^{2} and
cosx is the derivative of sinx. In this case, we say that x^{2} 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.

1) `int`e^{x } dx = e^{x}

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.

**Ex:1 int x ^{2} 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