7.4 Integration of Rational Functions by

7.4 Integration of Rational Functions by - 7.4 Integration...

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7.4 Integration of Rational Functions by Partial Fractions This section examines a procedure for decomposing a rational function into simpler rational functions to which we can apply the basic integration formulas. A. Partial Fraction Decomposition of Refer to the Handout of Integration of Rational Functions by Partial Fractions A. Rationalizing Substitutions B. Rational Functions of Sine and Cosine (Weierstrass Substitutions) ) ( ) ( x g x f
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A. Examples of Integrals of Rational Functions Distinct Linear Factors dx x x x x x - + + - + 2 0 2 2 3 12 1 12 Note: If the degree of the numerator is greater than or equal to the degree of the denominator, perform the long division first. Solution: 12 1 12 1 12 2 2 2 3 - + + = - + + - + x x x x x x x x where, 3 4 ) 3 )( 4 ( 1 12 1 2 - + + = - + = - + x B x A x x x x To find the values of A and B , multiply both sides by the lowest common denominator which leads to the basic equation. ) 4 ( ) 3 ( 1 + + - = x B x A Basic equation To solve for A , let in basic equation. We chose to eliminate the term 4 - = x 7 1 7 1 ) 4 4 ( ) 3 4 ( 1 - = - = + - + - - = A A B A
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