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.
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.
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.
The mathematical relationship between the indefinite integration and the differentiation can be understood from the following mathematical expression.
$\displaystyle \int{f(x)\,}dx \,=\, g(x)+c$ $\,\,\Longleftrightarrow\,\,$ $\dfrac{d}{dx}{\Big(g(x)+c\Big)}$ $\,=\,$ $f(x)$
Learn the list of indefinite integration formulas with mathematical proofs.
Learn how to use the integral rules in finding the indefinite integration of the functions.
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