The operation of integration, is the inverse of the operation of differentiation. For this reason, the term integral can also be referred to as antiderivative, a function F whose derivative is the given function f.

Notation for the Indefinite Integral

The ∫ sign is an elongated "S", standing for "sum". Later we will see that the integral is the sum of the areas of infinitesimally thin rectangles.
∑ is the symbol for "sum". It can be used for finite or infinite sums.
∫ is the symbol for the sum of an infinite number of infinitely small areas (or other variables).

Sometimes we write a capital letter to signify integration. For example, we write F(x) to mean the integral of f(x). So we have:
F(x) = ∫ f (x) dx
In this case, it is called an indefinite integral.


The integral sign ∫ represents integration. The symbol dx, called the differential of the variable x, indicates that the variable of integration is x. The function f(x) to be integrated is called the integrand. The symbol dx is separated from the integrand by a space (as shown).

Integrals are used extensively in many areas of mathematics as well as in many other areas that rely on mathematics.



Example 1
If we know that


and we need to know the function this derivative came from. Think: "What would I have to differentiate to get this result?"
and than...
we khow that the function 4x3 is result of the differentiation from x4

in other words, we know
y = x4 is one antiderivative of
and of course, There are infinitely many other antiderivatives which would also work, for example:
y = x4 + 2
y = x4 - (1/3)

In general, we say y = x4 + C is the indefinite integral of 4x3 . The number C is called the constant of integration.

Note: Most math text books use C for the constant of integration, but for questions involving electrical engineering, we prefer to write "K" for the constant of integration.


In example 1,
We can write: F(x) = ∫ f (x) dx = x4 + C and say:

"The integral of 4x3 with respect to x equals x4 + C."