Indefinite integration

The inverse mathematical operation of differentiation without defining the interval of a function is called the indefinite integration. It is also called as the antidifferentiation.

Introduction

Let $f(x)$ and $g(x)$ be two functions in terms of $x$.

A mathematical expression is defined as follows and it is formed by the differentiable function $g(x)$ and a mathematical constant $c$.

$g(x)+c$

The differentiation of this mathematical expression is written in the following mathematical form.

$\implies$ $\dfrac{d}{dx}{\Big(g(x)+c\Big)}$

Let’s assume that the derivative of the mathematical expression $g(x)+c$ is equal to the function $f(x)$.

$\implies$ $\dfrac{d}{dx}{\Big(g(x)+c\Big)}$ $\,=\,$ $f(x)$

Then, the inverse process of this mathematical operation is called integration or antidifferentiation, and it is expressed in calculus in the following form.

$\,\,\,\therefore\,\,\,\,\,\,$ $\displaystyle \int{f(x)\,}dx$ $\,=\,$ $g(x)+c$

Here, the symbol $\displaystyle \int{}$ is the integral symbol and the expression $\displaystyle \int{f(x)\,}dx$ is read as the integral of the function $f(x)$ with respect to $x$.

Actually, the integral of the function $f(x)$ with respect to $x$ is calculated without defining the interval of the function. Hence, this mathematical process is called the indefinite integration.

The function $g(x)$ is called by the following three ways.

A primitive of the function $f(x)$ with respect to $x$
An anti-derivative of the function $f(x)$ with respect to $x$
An indefinite integral of the function $f(x)$ with respect to $x$

In this case, the function $f(x)$ is called the integrand and the mathematical constant $c$ is called the constant of integration or integral constant.

Relationship

The mathematical relationship between the indefinite integration and the differentiation can be understood from the following mathematical expression.

$\displaystyle \int{f(x)\,}dx \,=\, g(x)+c$ $\,\,\Longleftrightarrow\,\,$ $\dfrac{d}{dx}{\Big(g(x)+c\Big)}$ $\,=\,$ $f(x)$