First you must factor the denominator 2 2 3 2 2 5 20

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First, you MUST factor the denominator. 2 2 3 2 2 5 20 6 5 20 6 2 ( 1) x x x x x x x x x + + + + = + + + And you can now set up the fractions: 2 2 2 5 20 6 ( 1) 1 ( 1) x x A B C x x x x x + + = + + + + + As in all previous examples, after this, you follow through with the common denominator, expansion, and grouping similar terms: 2 2 2 2 2 2 2 5 20 6 ( 1) ( 1) 5 20 6 ( 2 1) ( 1) 5 20 6 2 x x A x Bx x Cx x x A x x Bx x Cx x x Ax Ax A Bx Bx Cx + + = + + + + + + = + + + + + + + = + + + + + The 3 unknowns A, B, C can eventually be found: A = 6, B = -1 and C = 9 and from them, you can finish the integration. 2 2 1 3 2 2 2 5 20 6 5 20 6 6 1 9 6ln ln( 1) 9( 1) 2 ( 1) 1 ( 1) x x x x dx dx dx x x x C x x x x x x x x + + + + = = + = + + + + + + + + This example emphasizes that, you must factor the denominator first before applying the partial fraction.
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Phong Do Distinct Quadratic Factors 2 2 2 ? (3 1)( 1) x x dx x x + = + Your partial fraction set up is: Note the use of “ Bx + C ” for the quadratic factor. After setting the partial fractions, the rest will be just like before: Algebra to expand and group similar terms so that the corresponding coefficients can be equated to form a system of equations: You will find that: 7 4 3 5 5 5 A B C = = and from these, you can complete the integration:
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Phong Do Repeated Quadratic Factors Once you have a good grasp of the other cases, the case involving repeated quadratic factors are handled in exactly the same manner. The Algebra, even for simple problem, can be quite extensive. You first contruct the partial fractions: Per the partial fraction rule for repeated quadratic factors, you will end up with A, B, C, D and E as the unknowns. And you can guess that the Algebra will be very messy. After much work, you will have the system of equation shown below: And now, solving this system is not trivia and is best done on a computer (and of course, you may as well have the original integral solved using a computer algebra to begin with). To continue on, however, you will find that: And from which:
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Phong Do Today, complicated partial fractions are decomposed using computer. For example, the previous example is very quickly decomposed in Maplesoft as: And you did NOT even have to factor first: Improper Rational function revisited Just to make sure that you must perform long division (or synthetic division) to reduce your rational function to proper form, the next example should remind you. Because the degree of the numerator is larger, you must perform long division: The result of the division can now be shown as: Giving you: The first integral is just a straight power rule. And you need to use partial fraction in the last integral.
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Phong Do As with many other complex calculations today, much of the tedious work can easily be performed using the computer. If polynomial long divisoin is something that you want to avoid, tool such as MapleSoft can handle it nicely and quickly. Below are the “quotient” and “remainder” commands in MapleSoft for the example that you just saw: Some Comments on Partial Fractions Factor out the quadratic if possible When you encounter a problems containing quadratic factors in the denominator, sometimes it is beneficial to see if you could further factor the quadratic form to make the algebra a bit simpler.
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  • Summer '14
  • Calculus, Fractions, Fraction, Elementary algebra, Rational function, Partial fractions in integration

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