Chap7_Sec4

# Chap7_Sec4 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

This preview shows pages 1–19. Sign up to view the full content.

7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
7.4 Integration of Rational Functions by Partial Fractions TECHNIQUES OF INTEGRATION In this section, we will learn: How to integrate rational functions by reducing them to a sum of simpler fractions.
PARTIAL FRACTIONS We show how to integrate any rational function (a ratio of polynomials) by expressing it as a sum of simpler fractions, called partial fractions. We already know how to integrate partial functions.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
To illustrate the method, observe that, by taking the fractions 2/( x – 1) and 1/( x – 2) to a common denominator, we obtain: INTEGRATION BY PARTIAL FRACTIONS 2 2 1 2( 2) ( 1) 1 2 ( 1)( 2) 5 2 x x x x x x x x x + - - = = - + - + + = + -
If we now reverse the procedure, we see how to integrate the function on the right side of this equation: INTEGRATION BY PARTIAL FRACTIONS 2 5 2 1 2 1 2 2ln | 1| ln | 2 | x dx dx x x x x x x C + = - ÷ + - - + = - - + +

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
To see how the method of partial fractions works in general, let’s consider a rational function where P and Q are polynomials. INTEGRATION BY PARTIAL FRACTIONS ( ) ( ) ( ) P x f x Q x =
PROPER FUNCTION It’s possible to express f as a sum of simpler fractions if the degree of P is less than the degree of Q . Such a rational function is called proper.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Recall that, if where a n 0, then the degree of P is n and we write deg( P ) = n. DEGREE OF P 1 1 1 0 ( ) n n n n P x a x a x a x a - - = + +××× + +
If f is improper, that is, deg( P ) ≥ deg( Q ), then we must take the preliminary step of dividing Q into P (by long division). This is done until a remainder R ( x ) is obtained such that deg( R ) < deg( Q ). PARTIAL FRACTIONS

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The division statement is where S and R are also polynomials. PARTIAL FRACTIONS ( ) ( ) ( ) ( ) ( ) ( ) P x R x f x S x Q x Q x = = + Equation 1
PARTIAL FRACTIONS As the following example illustrates, sometimes, this preliminary step is all that is required.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Find The degree of the numerator is greater than that of the denominator. So, we first perform the long division. PARTIAL FRACTIONS Example 1 3 1 x x dx x + -
PARTIAL FRACTIONS This enables us to write: 3 2 3 2 2 2 1 1 2 2ln | 1| 3 2 x x dx x x dx x x x x x x C + = + + + ÷ - - = + + + - + Example 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The next step is to factor the denominator Q ( x ) as far as possible. PARTIAL FRACTIONS
FACTORISATION OF Q ( x ) It can be shown that any polynomial Q can be factored as a product of: Linear factors (of the form ax + b ) Irreducible quadratic factors (of the form ax 2 + bx + c , where b 2 – 4 ac < 0).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
FACTORISATION OF Q ( x ) For instance, if Q ( x ) = x 4 – 16, we could factor it as: 2 2 2 ( ) ( 4)( 4) ( 2)( 2)( 4) Q x x x x x x = - + = - + +
The third step is to express the proper rational function R ( x )/ Q ( x ) as a sum of partial fractions of the form: FACTORISATION OF Q ( x ) 2 or ( ) ( ) + + + + i j A Ax B ax b ax bx c

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
it is always possible to do this.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 89

Chap7_Sec4 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

This preview shows document pages 1 - 19. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online