Introduction
An antiderivative is a function that, when differentiated, gives the original function. In other words, it is the inverse operation of differentiation. Antiderivatives are essential in calculus and are used to find the area under a curve, the volume of a solid, and many other important quantities in mathematics, science, and engineering.
The process of finding antiderivatives is called integration, and it involves using various techniques such as substitution, integration by parts, and partial fractions. Antiderivatives are typically expressed with an indefinite integral, which contains a constant of integration that accounts for all possible antiderivatives of the function.
If f'(x) be the derivative of a function F(x) then the antiderivative of the function f(x) is defined as
$$\rm \int f(x) \ dx = F(x) + C$$
Here, $\rm \int$ is the integral symbol, f(x) is the function to be integrated, $\rm dx$ represents the infinitesimal change in the variable x, and F(x) is the antiderivative of f(x).
Antiderivatives have numerous applications in different fields. For example, in physics, antiderivatives are used to calculate the work done by a force, while in economics, they are used to determine the total revenue generated by a business.