Chap7_Sec4 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION
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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.
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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.
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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 + - - = = - + - + + = + -
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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 + = - ÷ + - - + = - - + +
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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 =
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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.
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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 - - = + +××× + +
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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
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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
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PARTIAL FRACTIONS As the following example illustrates, sometimes, this preliminary step is all that is required.
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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 + -
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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
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The next step is to factor the denominator Q ( x ) as far as possible. PARTIAL FRACTIONS
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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).
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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 = - + = - + +
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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
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it is always possible to do this.
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Chap7_Sec4 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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