Integral as Anti Derivative

Primitive or anti-derivative of a function:

A function ø is called a primitive or an anti 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).