MATH 1004 Lecture Notes - Lecture 2: Opata Language, Trigonometric Substitution, Sinar
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MATH 1004 Full Course Notes
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7. 4 partial fraction: 7. 4. 1 long division of polynomials p(x) q(x) p(x) q(x) Example: simplify such that deg p(x) deg q(x) = (new polynoimial) + remainder denominator x4 + 3x2 2x + 1 x + 1. Ap (x r)p: if the denominator has a quadratic irreducible factor of form (ax2 + bx + c)q, then partial fraction decomposition is form. B1x + c1 ax2 + bx + c. B2x + c2 (ax2 + bx + c)2 + + Bqx + cq (ax2 + bx + c)q: to nd as, bs and cs: Examples: decompose: f (x) = x 4 x(x 1)2 , f (x) = , f (x) = x3 + 2x2 1 (x 1)3(x3 1) 1: evaluate, evaluate (cid:90) 1 (cid:90) x2 + 2x + 1. 5x 7 x2 3x + 2 dx (cid:90) dx (cid:90) (cid:90) x4 + 1. 1 + x2 + 1 dx x dx x + 1 (x 1)2(x + 2)