23-Partial-Fraction - Integration using Partial Fraction...

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Phong Do Integration using Partial Fraction Partial fraction is a method allowing you to take a large fraction and “decompose” it into several single fractions. Working with these single fractions is often easier as the below example will show. Let’s say you have this rational expression : 1 ( 2)( 2) x x + From Pre-Cal (and if you forgot, we will do a quick review on Partial Fraction Decomposition) you can decompose this into 2 single fractions as: 1/ 4 1/ 4 1 ( 2)( 2) 2 2 x x x x = + + + If we take the above example and turn in into an integral, you will see how partial fraction will help you: 1/ 4 1/ 4 ( 2)( 2) 2 2 dx dx dx x x x x = + + + The 2 integrals containing the single fractions can be integrated as: 1 1 ln 2 ln 2 4 4 x x C + + + So as you can see, by splitting a single fraction into 2 smaller fractions each with single denominator, integration can be conveniently carried out. Proper and Improper Rational Functions: A Quick Pre-Cal Review Rational Functions has the form: ( ) ( ) P x Q x And in Algebra, you might have recalled that: A proper rational function is one where the power of the numerator is less than the power of the denominator An improper rational function is one where the power of the numerator is greater than or equal to the power of the denominator . Partial Fraction technique requres a rational function to be PROPER . 3 Proper ( 2)( 4) x x x + 3 4 2 2 3 Proper 3 5 x x x x + 3 2 2 3 Improper 4 7 1 x x x x + + + + If a rational expression is improper, you use polynomial long division (or synthetic division if appropriate) to reduce it to a polynomial + a proper rational function ( ) 3 2 2 2 3 4 7 3 2 3 4 1 1 x x x x x x x + + + + + + + The 2 3 2 1 x x + is proper which you can use Partial Fraction on.
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