Partial Fractions Method in Integration, Study notes of Mathematics

Download Partial Fractions Method in Integration and more Study notes Mathematics in PDF only on Docsity! 7.3 Partial Fractions 1 Chapter 7. Integration Techniques, l’Hôpital’s Rule, and Improper Integrals 7.3 Partial Fractions Note. Consider ∫ 1 1− x2 dx. We can see that we have the algebraic identity 1 1− x2 = 1/2 1 + x + 1/2 1− x. Then we have ∫ 1 1− x2 dx = ∫ 1/2 1 + x dx + 1/2 1− x dx = 1 2 ln |1 + x| − 1 2 ln |1− x| = 1 2 ln ∣∣∣∣1 + x1− x ∣∣∣∣ = 1 2 ln 1 + x 1− x if |x| < 1 = tanh−1 x if |x| < 1 What simplified this computation was breaking up the denominator and “undoing” the common denominator process. This will be the idea of the method of this section, which is called the method of partial fractions. 7.3 Partial Fractions 2 Note. A polynomial with real coefficients can be factored into linear factors (x − ri) and irreducible quadratics (x2 + pjx + qj). To show this, we need to know some complex variables (this result is presented in our Complex Variables class, MATH 4337/5337). Note. Method of Partial Fractions. If f and g are polynomials, to integrate f/g: 1. If the degree of f is greater than or equal to the degree of g, perform long division. 2. Factor g into linear factors (x − r) and irreducible quadratics (x2 + px + q). 3. For each linear factor (x − r) of g of order m (that is, (x − r) divides g m times), take the partial fractions A1 (x − r) + A2 (x − r)2 + · · · + Am (x − r)m. 4. For each irreducible quadratic factor x2 + px + q of g of order n, take the partial fractions B1x + C1 (x2 + px + q) + B2x + C2 (x2 + px + q)2 + · · · + Bnx + Cn (x2 + px + q)n . 5. Set f/g equal to the sum of all partial fractions.