Partial Fractions Decomposition: A Method for Antidifferentiating Rational Expressions, Study notes of Algebra

Download Partial Fractions Decomposition: A Method for Antidifferentiating Rational Expressions and more Study notes Algebra in PDF only on Docsity! Partial Fractions Reference Sheet The method of partial fraction decomposition (PFD) allows us to antidifferentiate rational expressions by rewriting them in terms of simpler rational expressions that we know how to antidifferentiate. Recall from algebra: Every polynomial with real coefficients can be factored into linear and irreducible quadratic factors. A typical problem might look like this: ! 2𝑥$ − 4𝑥 − 8 (𝑥) − 𝑥)(𝑥) + 4) 𝑑𝑥 Partial Fraction Decomposition – The Method that Always Works 1. If the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator, use long division or synthetic division to rewrite the integrand. 2. Then factor the denominator into linear factors and irreducible quadratic factors. 3. Repeated Linear Factors: For each factor of the form (𝑚𝑥 + 𝑏)/ , the partial fraction decomposition must include the following terms: 𝐴 𝑚𝑥 + 𝑏 , 𝐵 (𝑚𝑥 + 𝑏)) , 𝐶 (𝑚𝑥 + 𝑏)$ , … , 𝑁 (𝑚𝑥 + 𝑏)/ 4. Repeated Quadratic Factors: For each factor of the form (𝑎𝑥) + 𝑏𝑥 + 𝑐)/, where 𝑏) − 4𝑎𝑐 < 0, the partial fraction decomposition must include the following terms: 𝐴𝑥 + 𝐵 𝑎𝑥) + 𝑏𝑥 + 𝑐 , 𝐶𝑥 + 𝐷 (𝑎𝑥) + 𝑏𝑥 + 𝑐)) , 𝐸𝑥 + 𝐹 (𝑎𝑥) + 𝑏𝑥 + 𝑐)$ , … , 𝐺𝑥 + 𝐻 (𝑎𝑥) + 𝑏𝑥 + 𝑐)/ 5. Then add the terms from steps (3) and (4) together. This is form of the partial fraction decomposition of the original rational expression. 6. Multiply the form from (5) by the LCD, and collect like terms in 𝑥/, 𝑥/?@,… , 𝑥), 𝑥, and so on. 7. Then equate the coefficients of like terms on both sides. We can do this because two polynomials are equal if and only if their coefficients are equal. 8. Solve the resulting system of equations for the constants 𝐴, 𝐵, 𝐶, etc. from the form in (5). Substitute these constants into that form. 9. Now the rational expression may be antidifferentiated using basic rules, yielding expressions whose antiderivatives are natural logarithmic functions, power functions requiring 𝑢- substitution, and often arctangent functions. There are short-cuts for linear factors, as described in the detailed lesson notes.