Integration of Polynomials

algebra.jpgIntegration means finding the area between the graph of a function and the x-axis.

The process of integration can be thought of as the reverse of differentiation. It is also known as anti-differentiation.

Given a derived function, integrating will revert back to the original function.

The result of integration is called the integral or anti-derivative.

e.g If x2 is differentiated the result is 2x

The anti-derivative of 2x is x2.

The problem is that many other functions differentiate to 2x. e.g. x2 + 6, x2 − 20 etc.

To allow for this possibility a constant of integration(c) or arbitrary constant has to be added each time we integrate.

This type of integral is called an indefinite integral because the value of c is unknown.

The definite integral is a number, the indefinite is a function.

In general, the rule for integrating powers of x (with the exception of x -1) is "add one to the exponent and divide by the new exponent."

The symbol used to indicate integration is ∫

e.g ∫ 2x dx means "integrate 2x with respect to x" The dx part is simply to indicate that x is the variable.

Methods of Integration

The formula for integration is:

∫ xn dx = Y12_Integration_of_Polynomials_01.gif + c

For a term with a coefficient not equal to 1, the formula becomes:

∫ axn dx =a. Y12_Integration_of_Polynomials_02.gif + c

This means that a coefficient remains unchanged by integration.

Functions containing all types of terms and expressions can be integrated. e.g. Trigonometric, logarithmic, roots etc.

Anti-differentiating can be explained by asking "What must I differentiate to get this function"

Notes

Polynomials are integrated term by term.

 (see Example 2)

Expressions with brackets should usually be expanded first.

 (see Example 3)

In rational functions common factors should be cancelled

 (see Example 4)

 

 
Examples
Answers
1 Find ∫ x3 dx ∫ x3 dx = Y12_Integration_of_Polynomials_03.gif+ c
2 Find ∫ (3x2 + 4x + 5) dx
∫ (3x2 + 4x + 5) dx=
 Y12_Integration_of_Polynomials_04.gif+c
=
 x3 + 2x2 + 5x + c
3 Find ∫ (x − 2)(x + 4) dx
∫ (x − 2)(x + 4) dx =
 ∫ x2 + 2x − 8 dx
=

Y12_Integration_of_Polynomials_05.gif+ c
=
 x3/3 + x2 − 8x + c
4 Find Y12_Integration_of_Polynomials_06.gif Y12_Integration_of_Polynomials_07.gif

 

Finding the Constant of Integration

Sometimes enough information is given to enable the value of the constant to be found.

Example

Anti-differentiate to help find f(x) if f '(x) = 4x + 2 and f(1) = 8

∫ 4x + 2 dx = 2x2 + 2x + c

i.e. f(x) = 2x2 + 2x + c

8 = 2 x 12 + 2 × 1 + c

8 = 4 + c

c = 4

f(x) = 2x2 + 2x + 4