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Transcript
Warm-Up: Common Denominators Notes Write the sum of rational functions as a single rational function, by finding a common denominator: x +1 2 + =? x2 + 4 x + 3 Notes Techniques of Integration: Partial Fraction Decomposition (sec 7.5) 22 September 2014 Today, We Will: I Learn to apply the method of partial fraction decomposition to re-write a rational function as a sum of rational functions whose denominators are polynomials of degree less than or equal to two. Notes Terminology Notes Recall: I A rational function is a function that is the quotient of two polynomials. I The degree of a polynomial is the highest power of a variable occurring in the polynomial. I A polynomial of degree one is called a linear polynomial. I A polynomial of degree two is called a quadratic polynomial. I A quadratic polynomial that cannot be factored into linear polynomials is called irreducible . Motivation Notes Consider the integral: Z 3x 2 + 4x + 11 dx. x 3 + 3x 2 + 4x + 12 Problem: Except in rare situations where a u-substitution is possible, we don’t have any integration techniques that let us deal with rational functions if the degree of the polynomial in the denominator is greater than two. continued . . . Motivation, Continued Notes From the warm-up, however, we know that: Z 3x 2 + 4x + 11 dx = x 3 + 3x 2 + 4x + 12 = These we can integrate! Z x +1 2 + dx x2 + 4 x + 3 Z x dx + x2 + 4 Z 1 dx + x2 + 4 Z 2 dx x +3 Partial Fraction Decomposition Notes Partial fraction decomposition is an algebraic technique that “undoes” the process of combining rational functions by finding a common denominator. It allows us to take a rational function whose denominator is a polynomial of degree greater than two, and write it as a sum of rational functions whose denominators are linear and quadratic polynomials. Once we have found the decomposition of the original rational function, we can apply a combination of techniques — usually u-substitution and trigonometric substitution — to evaluate the integral. Examples: Partial Fraction Decomposition I R(x) = x +2 x2 + x − 6 I R(x) = x +2 (x + 3)(x − 2)2 The Method Notes Notes I Make sure the degree of the denominator is greater than the degree of the numerator. If it is not, divide the numerator by the denominator to re-write the original function as the sum of a polynomial and a rational function. I Factor the denominator into powers of linear and irreducible quadratic polynomials. continued . . . The Method, Continued I Notes For every power of a linear factor (ax + b)n , write the sum: A1 A2 An + + ··· + ax + b (ax + b)2 (ax + b)n where A1 , A2 , . . . , A3 are unknown constants. continued . . . The Method, Continued I Notes For every power of an irreducible quadratic factor (ax 2 + bx + c)n , write the sum: A1 x + B1 A2 x + B2 An x + Bn +···+ + ax 2 + bx + c (ax 2 + bx + c)2 (ax 2 + bx + c)n where A1 , A2 , . . . , An and B1 , B2 , . . . , Bn are unknown constants. continued . . . The Method, Continued I Combine all resulting sums into a single rational function. I Clear the denominators and equate the numerator of the resulting rational function and the numerator of the original rational function. I Solve for the unknown constants. Notes