Integration

Definition

An Antiderivative or indefinite integral of a function f is a differentiable function F whose derivative is equal to f, i.e. F'=f

The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

Example

The function F(x) = \dfrac{x^3}{3} is an antiderivative of f(x) = x^2. As the derivative of a constant is zero, x2 will have an infinite number of antiderivatives; such as (\dfrac{x^3}{3}) + 0, (\dfrac{x^3}{3}) + 7, (\dfrac{x^3}{3}) − 42, (\dfrac{x^3}{3}) + 293 etc.

Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F(x) = \dfrac{x^3}{3} + C; where C is an arbitrary constant known as the constant of integration.

Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph’s vertical location depending upon the value of C.

Notation

The simplest case, the integral over x of a real-valued function f(x), is written as \int f(x)\ dx. The integral sign represents integration. The dx indicates that we are integrating over x; x is called the variable of integration.