Primitive or anti-derivative of a function:
A function ø is called a primitive or an ant derivative of a function f(x) if ø’(x) = f(x).
Let ø (x) be a primitive of a function f(x) and let C be any constant then,
d/dx [ø (x) + C] = ø’ (x) = f (x) [∵ø’ (x) = f (x)]
⇒ ø (x) + C is also a primitive of f(x)
Thus, if a function f(x) posses a primitive, then it possess infinity many primitives which are contained in the expression ø (x) + C, where C is a constant.
Indefinite integral or indefinite integration:
Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by ∫f(x) dx.
Thus, d/dx [ø’ (x) + C] = f (x) ⇔ ∫f(x) dx = ø (x) + C … (i)
Where ø (x) is primitive of f(x) and C is an arbitrary constant known as the constant of integration.
Here ∫ is the integral sign f(x) is the integral, x is the variable of integration and dx is the element of integration or differential of x.
The process of finding an indefinite integral of a given function is called integration of the following.
It follows from the above discussion that of a given function is called integration of the function.
It follows from the above discussion that integrating a function f(x) means finding a function ø (x) such that d/dx (ø(x)) = f (x).