sec7_4

sec7_4 - A 1 x + B 1 ( ax 2 + bx + c ) + A 2 x + B 2 ( ax 2...

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7.4 Integration of Rational Functions by Partial Fractions 1. Background Remark 1.1. This method should only be used when a method from calculus I will not work. Example 1.1. Use Calc I methods: ± x 2 +3 x - 2 x dx Example 1.2. Use Calc I methods: ± x x 2 +1 dx Example 1.3. Can use either trigonometric substitution or partial fraction decom- position: ± 1 x 2 - 25 dx 2. Decomposing Rational Functions Example 2.1. Decompose x 2 +8 x 2 - 5 x +6 Step 1. Divide if the degree of the numerator is greater than or equal to the degree of the denominator. i.e. rewrite as follows f ( x ) g ( x ) = Q ( x )+ R ( x ) g ( x ) , deg ( R ( x )) < deg ( g ( x )) Step 2. Completely factor the denominator into linear and quadratic factors: ( ax + b ) n and/or ( ax 2 + bx + c ) m Step 3. For each distinct linear factor, ( ax + b ) n , the partial fraction decomposition will include the sum A 1 ( ax + b ) + A 2 ( ax + b ) 2 + A 3 ( ax + b ) 3 + ··· + A n ( ax + b ) n Step 4. For each distinct quadratic factor, ( ax 2 + bx + c ) m , the partial fraction de- composition will include the sum
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Unformatted text preview: A 1 x + B 1 ( ax 2 + bx + c ) + A 2 x + B 2 ( ax 2 + bx + c ) 2 + A 3 x + B 3 ( ax 2 + bx + c ) 3 + ··· + A m x + B m ( ax 2 + bx + c ) m 1 Section 7.4 Partial Fractions 2 Example 2.2. Evaluate ± x 2 + 8 x 2-5 x + 6 dx Example 2.3. Decompose 3 x 3-x 2 + 6 x-4 ( x 2 + 1)( x 2 + 2) Example 2.4. Evaluate ± 3 x 3-x 2 + 6 x-4 ( x 2 + 1)( x 2 + 2) dx Section 7.4 Partial Fractions 3 3. Integrating by Partial Fractions Example 3.1. ± 2 1 1 x 2 + 2 x dx Example 3.2. ± cos x sin 2 x + sin x-6 dx Example 3.3. ± x 3 x 2 + 2 x + 1 dx Section 7.4 Partial Fractions 4 4. Rationalizing Substitution Make a substitution of the form u = n ± f ( x ) to change the integrand into a rational function. Example 4.1. ² 16 9 √ x x-4 dx Example 4.2. ² e 2 x e 2 x + 3 e x + 2 dx...
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This note was uploaded on 03/14/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

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sec7_4 - A 1 x + B 1 ( ax 2 + bx + c ) + A 2 x + B 2 ( ax 2...

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