If y = f(x), then `dy/dx` =f'(x)
Given a function f(x), we have seen the process of differentiation of f(x) wrt x. If f'(x) is given, how to find f(x). For example, for which function, the derivative is 3x^{2}? We know that d/dx (x^{3}) is 3x^{2} . Therefore, x3 is the function for which derivative is 3x^{2.}
Similarly d/dx(logx) = 1/x . i.e. the derivative of log x wrt x is 1/x. As a converse, we sat that the integral of 1/x is log x.
Hence integration is the anti derivative process. It is the reverse process of differentiation.
Definition: A function F(x) is called an anti derivative or integral of a function f(x) on an interval I if F'(x) = f(x) for every value of x in I,
i.e. If the derivative of a function F(x) wrt x is f(x), then we say that the integral of f(x) wrt x is F(x).
i.e. `int` f(x) dx = F(x).
For example, we know that d/dx (sinx) = cos x, then `int` cos x dx = sin x.
Also, d/dx (x^{5}) = 5x^{4}, gives `int` 5x^{4}dx =x^{5}
The symbol `int` is the sign of integration. The symbol of integration is elongated S, which is the first letter of the word "Sum".
The function f(x) is called the integrand. The variable x in dx is called variable of integration or integrator. The process of finding the integral is called integration.
1. d/dx ((x+1) =1
d/dx(x) =1
d/dx (x3) =1. So we come to conclusion that `int` dx = x+C where C can be any constant.
So it is accepted to understand that `int` f(x) dx is not a particular integral, but a family of integrals of that function.
If F(x) is one such integral, it is customary to write `int` f(x) dx = F(x) +C, where C is arbitrary constat. "C" is called the constant of integration. Since C is arbitrary, `int` f(x) dx is called the 'indefinite integral'.
Some important formulae in finding antiderivatives:
`int` x^{n}dx = x^{n+1}/n+1 +c (n `!=` 1)  `int` cosxdx = sinx +c 
`int` 1/x^{n} dx = 1/(n1)*x^{n1} +c 
`int` cosec^{2}x = cotx+c

`int` 1/x dx = log x+c  `int` sec^{2}x = tanx +c 
`int` e^{x}dx = e^{x} +c  `int` sec x tanxdx = sec x +c 
`int` a^{x}dx = a^{x}/loga +c  `int` csc x cot x dx = cotx +c 
`int` sinxdx=cosx+c  `int` 1/1+x^{2}dx = tan‾x+c 
`int` 1/x`sqrt(x21)` dx = sec‾x+c  `int` 1/`sqrt(1x2)` dx = arcsinx+c 
These are the formulae which can be directly applied to find out integrals of standard functions.
Problems on solving antiderivative of a function:
1. Find integral of i) x^{16} ii) x^{5/2} iii)1/x^{5} iv) 1/csc x dx v) cosx/sin^{2}x
i) `int` x^{16}dx = x^{16+1}/16+1 = x^{17}/17+c = `(x17)/(17)` +c
ii) `int` x^{5/2} dx = x ^{5/2+1}/5/2+1 = x^{7/2}/7/2+c = `(2x7/2)/7)`
iii) `int` 1/x^{5} dx = x^{5} dx = x^{5+1}/5+1 +c = x^{4}/4+c = 1/4x^{4} +c
iv) `int` 1/cscx dx = sinx dx = cosx
v) `int` cosx/sin^{2}x = cosx/sinx *1/sinx dx = cotxcosec x dx = cosecx+c
Conclusion: In this article, we studied about antiderivative of a function, applications, important formule, etc,