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• Indefinite integral calculus

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.

1. A primitive of the function $f(x)$ with respect to $x$
2. An anti-derivative of the function $f(x)$ with respect to $x$
3. 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.

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

Integral rules

Learn the list of indefinite integration formulas with mathematical proofs.

Problems

Learn how to use the integral rules in finding the indefinite integration of the functions.

Ashok Kumar, B.E.

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A Specialist in Mathematics, Physics, and Engineering with 14 years of experience helping students master complex concepts from basics to advanced levels with clarity and precision.

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