Essentially, integration is considered to be the reverse process of differentiation. Recall that once an expression is differentiated we obtain its derivative. Similarly, once an expression is integrated we obtain what is known as its integral. An integral is also sometimes referred to as an antiderivative, especially when discussing the Fundamental Theorem of Calculus.

The Fundamental Theorem of Calculus – Part 1

The Fundamental Theorem of Calculus is split into two parts. The formal definitions are too advanced for this page but we will explore what the theorem states loosely. The first part of the theorem, often termed the First Fundamental Theorem of Calculus, relates to indefinite integration.

Indefinite Integration

Consider differentiating y=x^2 to get \frac{dy}{dx}=2x. This derivative is also obtained when differentiating y=x^2+2, y=x^2-5 or y=x^2+c where c is ANY constant. When differentiating, constants disappear. Integration can be considered as the reverse process to differentiation and it is not possible to retrieve the original constant without further information. Hence, when integrating we add on a generic constant known as the constant of integration. This is known as indefinite integration. See Integrating Polynomials below for more information.

The First Fundamental Theorem of Calculus loosely states that, for continuous functions f(x), the indefinite integral F(x) (or antiderivative) of f(x) exists. It is found through integration – the reverse process to differentiation, i.e. f(x)=F'(x). See Definite Integrals for the second part of the theorem.

Given any expression, say y, we write its indefinite integral as

\int y\,dx

It is often read as ‘the integral of y with respect to x’.  Definite integrals have limits – numbers located on the integral sign. See Definite Integrals.


Integrating Polynomials

In order to integrate a polynomial, first recall how a polynomial is differentiated.

Differentiating a polynomial term requires first multiplying down by the power then reducing the power by one. If integration is the reverse process, then integrating a polynomial term can be obtained by increasing the power by one then dividing by the new power.

For example, consider y=x^3, differentiating gives \frac{dy}{dx}=3x^2. Adding one to the power of this and then dividing by the new power gets us back to the original expression: y=\frac{3x^3}{3}+c=x^3+c, where we have included the constant of integration as mentioned above.

Since integrating a polynomial term requires adding one to the power then dividing by the new power, we can use integral notation to express this:

\int x^n\, dx=\frac{1}{n+1}x^{n+1}+c

for n\ne -1 and where c is the constant of integration. This constant can be found if additional information is provided such as the coordinates of a point. See Example 2.

Just like when differentiating, when integrating, you may be required to write the expression as a polynomial first. This can also be seen in Example 2.


Integration Examples

Example 1

  1. Find y if \frac{dy}{dx}=6x^2-3x-4
  2. If f, find f(x).
  3. Find \int y\,dx if y=(2x-3)^2

Example 2

The curve C with equation y=f(x) passes through the point (1,2). Given that

f

find the equation of C.


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