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Lesson 7-4

Lesson 7-4. Partial Fractions. Fractional Integral Types. Type I – Improper: Degree of numerator ≥ degree of denominator Start with long division Type II – Proper: Degree of numerator < degree of denominator Check common forms or decompose into partial fractions

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Lesson 7-4

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  1. Lesson 7-4 Partial Fractions

  2. Fractional Integral Types • Type I – Improper: • Degree of numerator ≥ degree of denominator • Start with long division • Type II – Proper: • Degree of numerator < degree of denominator • Check common forms or decompose into partial fractions • Repeated terms require special techniques • Type III – Variations of Arctan: • U-substitution used a lot • Type IV – Variations of Arcsin: • U-substitution used a lot

  3. Strategies for Fractional Integrals

  4. 7-4 Example 1 1 x-1x + 0 x - 1 + 1 x --------- dx = x - 1  Long Division 1 = 1 dx + --------- dx x - 1   = x + ln |x – 1| + C

  5. 7-4 Example 2 x² - x + 1 x+1x³+ 0x² + 0x + 0 x³ + x² -x² + 0x -x² - x x + 0 x + 1 -1 x³ --------- dx = x + 1  Long Division 1 = x² dx - x dx + 1 dx - --------- dx x + 1     = ⅓x³ - ½x² + x - ln |x+1| + C

  6. Partial Fractions Example 3x – 1 Example: -------------- dx x² - x – 6 ∫ • factor denominator • rewrite fraction • multiply through by common denominator • solve one factor and substitute • repeat with remaining factor(s • substitute A and B and integrate x² - x – 6 = (x – 3) (x – 2) 3x – 1 A B ------------- = --------- + ---------- x² – x – 6 (x – 3) (x – 2) 3x – 1 = A(x + 2) + B(x – 3) when x = -2 then B = 7/5 when x = 3 then A = 8/5 8 1 7 1 -- -------- dx + -- --------- dx 5 (x-3) 5 (x + 2) = (8/5) ln|x-3| + (7/5) ln|x+2| + C ∫ ∫

  7. 7-4 Example 4 x + 7 -------------- dx = x² - x - 6 x + 7 ------------------ dx (x - 3)(x + 2) A B ---------- dx + ---------- dx (x - 3) (x + 2) 2 1 ---------- dx - ---------- dx (x - 3) (x + 2)       B(x - 3) + A(x + 2) = x + 7 B(-5) = -2 + 7 -5B = 5 B = -1 A(5) = 3 + 7 5A = 10 A = 2 2 ln|x-3| - ln|x+2| + C

  8. 7-4 Example 5 5x² + 20x + 6 --------------------- dx = x³ + 2x² + x 5x² + 20x + 6 ---------------------- dx x(x + 1)(x + 1) A B C ------ dx + ---------- dx + ----------- dx (x) (x +1) (x + 1)²      A(x +1)² + B(x)(x + 1) + Cx = 5x² + 20x + 6 A + B = 5 (x²) 2A + B + C = 20 (x) A = 6 (#) 6 + B = 5 B = -1 2(6) + -1 + C = 20 C = 9 6 1 9 ---- dx - ---------- dx + ------------ dx x (x + 1) (x + 1)²    6 ln|x| - ln|x+1| - 9/(x+1) + C

  9. 7-4 Example 6 k k u Variations of Arctan: ---------- du = --- tan-1 (---) + C u² + a² a a  dx -------------- 4x² + 9 1 2x ---- tan-1 (------) + C 6 3  u = 2x and a = 3 dx -------------- (x+1)² + 4 1 x+1 ---- tan-1 (--------) + C 2 2  u = x+1 and a = 2 dx ---------------- x² + 4x + 5 1 x+2 ---- tan-1 (--------) + C 1 1  x² + 4x + 4 + 1 = (x+2)² + 1² u = x+2 and a = 1

  10. 7-4 Example 7 k k x Variations of Arcsin: ---------- dx = --- sin-1 (---) + C a² - x² a a  dx -------------- 9 - x² 1 x ---- sin-1 (------) + C 3 3  u = x and a = 3 dx -------------- 16 – (1+x)² 1 x+1 ---- sin-1 (--------) + C 4 4  u = x+1 and a = 4 dx ----------------  -4x - x² 1 x+2 ---- sin-1 (--------) + C 2 2  4 – 4 – 4x - x² = 2² - (x+2)² u = x+2 and a = 2

  11. Summary & Homework • Summary: • No quotient rule for integration • Compare highest power numerator vs denominator • Numerator  Denominator • Use long division to simplify • Numerator < Denominator • Find common form • Use partial fractions • Homework: • pg 504-505, Day 1: 1, 2, 3, 7 Day 2: 4, 10, 19, 40

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