1 / 11

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

faraji
Download Presentation

Lesson 7-4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related