Anti-differentiation
Integration is an inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its original function. Such a process is called integration or anti-differentiation.
Therefore, the primary object of differential calculus is: 'given a function, to find its differential coefficient' whereas the primary object of integral calculus is its inverse, i.e., 'given the differential coefficient of a function, to find the function itself'.
Definition:
Let f(x) be the given function of 'x'. The family of all anti-derivatives of a function f(x) is called the indefinite integral of f with respect to x and is denoted by f(x) dx.
If φ(x) is any function such that φ'(x) = f(x),
then f(x) dx = φ(x) + c
where 'c' is an arbitrary constant, called the constant of integration.
In the notation f(x) dx,
the symbol was introduced by Leibniz and is called the integral sign,
the function to be integrated, i.e., f(x) is called the integrand
and 'dx' indicates that 'x' is the variable of integration.