mth.122.handout.04

mth.122.handout.04 - MTH 122 Calculus II Essex County...

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MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #4 — Sakai Web Project Material 1 Integration of Rational Functions Using Partial Fractions In the past it was far more usual to simplify or combine fractions than it was to expand them. However, it is sometimes easier to break an integrand up into simpler problems, and for rational expressions we have several forms to recognize. And I want to again emphasize that recognition does not necessarily mean memorization . Here’s a partial list that should be recognized, and I am putting a boxes around those that may be helpful when integrating rational functions: 1. d d x ( sinh - 1 x ) = 1 1 + x 2 . Which of course leads to: Z 1 1 + x 2 d x = sinh - 1 x + C 2. d d x ( cosh - 1 x ) = 1 x 2 - 1 . Which of course leads to: Z 1 x 2 - 1 d x = cosh - 1 x + C 3. d d x ( tanh - 1 x ) = 1 1 - x 2 . Which of course leads to: Z 1 1 - x 2 d x = tanh - 1 x + C 4. d d x ( csch - 1 x ) = - 1 | x | 1 + x 2 . Which of course leads to: Z 1 | x | 1 + x 2 d x = - csch - 1 x + C 5. d d x ( sech - 1 x ) = - 1 x 1 - x 2 . Which of course leads to: Z 1 x 1 - x 2 d x = - sech - 1 x + C 6. d d x ( coth - 1 x ) = 1 1 - x 2 . Which of course leads to: Z 1 1 - x 2 d x = coth - 1 x + C 1 This document was prepared by Ron Bannon ( ron.bannon@mathography.org ) using L A T E X2 ε . Last revised September 8, 2009. 1
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I think, however, that these should probably be memorized, at least for an exam. 1. Z 1 1 - x 2 d x = arcsin x + C 2. Z 1 1 + x 2 d x = arctan x + C 3. Z 1 x d x = ln | x | + C 2 The Pre-Calulus of Partial Fraction Decomposition You’ll have a rational function in lowest terms of the form P ( x ) Q ( x ) . Then you’ll need to follow the steps below. 1. If the degree of the numerator is greater than or equal to the denominator, long divide. You will be left with a rational function. For example: 6 x 3 + 5 x 2 - 7 3 x 2 - 2 x - 1 = 2 x + 3 + 8 x - 4 3 x 2 - 2 x - 1 . 2. If the degree of the numerator is less than the degree of the denominator, then factor the denominator into linear factors or irreducible 2 quadratic factors. An irreducible quadratic factor is a quadratic that cannot not be factored into linear factors with real coefficients. Example continued from above.
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mth.122.handout.04 - MTH 122 Calculus II Essex County...

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