sec7_4

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

This preview shows pages 1–4. Sign up to view the full content.

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

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 03/14/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

Page1 / 4

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online