Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Office Hours 9:15-10:15 MTWThF Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Monday, February 18 7.6 Strategies for integration page 421: 44, 46, 48, 50 Previous Assignment: Page 410 Problem 6 x - 7 INT --------------- dx 2 x - x - 12 x - 7 A B --------------- = ---------- + ---------- (x+3)(x-4) x+3 x-4 x - 7 = A(x-4) + B(x+3) = (A+B)x + (-4A+3B) A B 1 1 1 -4 3 -7 1 1 1 0 +7 -3 1 1 1 0 1 -3/7 1 0 10/7 0 1 -3/7 x - 7 10/7 -3/7 --------------- = ---------- + ---------- (x+3)(x-4) x+3 x-4 x - 7 INT ------------ dx = 10/7 ln(x+3) - 3/7 ln(x-4) + C (x+3)(x-4) ................................................................ Page 410 Problem 10 2 2 x - x - 20 INT ------------------- dx 2 x + x - 6 2 ------------------ 2 | 2 x + x - 6 | 2 x - x - 20 2 2 x + 2x - 12 ------------------- -3x - 8 2 2 x - x - 20 -3 x - 8 ------------------- = 2 + --------------- 2 (x+3)(x-2) x + x - 6 2 2 x - x - 20 A B ------------------- = 2 + ------ + ------ 2 x+3 x-2 x + x - 6 -3 x - 8 = A(x-2) + B(x+3) -3 x - 8 = (A+B)x +(-2A+3B) A B RHS 1 1 -3 -2 3 -8 1 1 -3 0 5 -14 1 1 -3 0 1 -14/5 1 0 -1/5 0 1 -14/5 2 2 x - x - 20 -1/5 -14/5 ------------------- = 2 + ------ + ------ 2 x+3 x-2 x + x - 6 2 2 x - x - 20 INT ---------------- dx = 2x - 1/5 ln(x+3) - 14/5 ln(x-2) + C 2 x + x - 6 --------------------------------------------------------------------- Page 410 Problem 14 2 7x + 2 x - 3 INT ---------------- dx (2x-1)(3x+2)(x-3) 2 7x + 2 x - 3 A B C ---------------- = ---------- + -------------- + ----------- (2x-1)(3x+2)(x-3) 2x-1 3x+2 x-3 2 7 x + 2 x - 3 = A(3x+2)(x-3) + B(2x-1)(x-3) + C(2x-1)(3x+2) 2 2 2 = A(3x - 7 x - 6) + B( 2x - 7 x + 3) + C(6x + x - 2) A B C RHS 3 2 6 7 -7 -7 1 2 -6 3 -2 -3 3 2 6 7 -1 -3 13 16 0 7 10 11 0 -7 45 55 1 3 -13 -16 0 7 10 11 1 3 -13 -16 0 7 10 11 0 -7 45 55 1 3 -13 -16 0 7 10 11 0 0 55 66 1 3 -13 -16 0 7 10 11 0 0 1 6/5 1 3 0 -2/5 0 7 0 -1 0 0 1 6/5 1 3 0 -2/5 0 1 0 -1/7 0 0 1 6/5 1 0 0 -2/5+3/7 0 1 0 -1/7 0 0 1 6/5 1 0 0 1/35 0 1 0 -1/7 0 0 1 6/5 2 7x + 2 x - 3 1/35 -1/7 6/5 INT ---------------- dx = INT ---------- + ---------- + -------- dx (2x-1)(3x+2)(x-3) 2x-1 3x+2 x-3 2 7x + 2 x - 3 INT ---------------- dx = 1/35 ln(2x-1)/2 - 1/7 ln(3x+2)/3 + 6/5 ln(x-3) + C (2x-1)(3x+2)(x-3) ------------------------------------------------------------------------- Page 410 Problem 20 6 3 x + 4 x + 4 INT -------------------- dx 3 2 x - 4 x 3 2 x + 4 x + 16 x + 68 ------------------------------ 3 2 | 6 3 x - 4 x | x + 4 x + 4 6 5 x - 4 x ------------------------- 5 4 x 5 4 4 x - 16 x ------------------------ 4 16 x 4 3 16 x - 64 x -------------------- 3 68 x 3 2 68 x - 272 x ----------------------- 2 272 x + 4 6 3 2 x + 4 x + 4 3 2 272 x + 4 INT -------------------- dx = x + 4 x + 16 x + 68 + -------------- 3 2 2 x - 4 x x ( x-4 ) 6 3 x + 4 x + 4 3 2 A B C INT ----------------- dx = x + 4 x + 16 x + 68 + --- + ------ + ---- 3 2 x 2 x-4 x - 4 x x 2 2 272 x + 4 = A x(x-4) + B(x-4) + C x 2 2 2 272 x + 4 = A( x - 4 x) + B(x-4) + C x A B C 1 0 1 | 272 -4 1 0 | 0 0 -4 0 | 4 1 0 1 | 272 -4 1 0 | 0 0 1 0 | -1 1 0 1 | 272 -4 0 0 | 1 0 1 0 | -1 1 0 1 | 272 1 0 0 | -1/4 0 1 0 | -1 1 0 0 | -1/4 0 1 0 | -1 1 0 1 | 272 1 0 0 | -1/4 0 1 0 | -1 0 0 1 | 272+1/4 1 0 0 | -1/4 0 1 0 | -1 0 0 1 | 1089/4 6 3 x + 4 x + 4 3 2 -1/4 -1 1089/4 INT --------------- dx = INT x + 4 x + 16 x + 68 + --- + ------ + ---- dx 3 2 x 2 x-4 x - 4 x x 6 3 4 3 2 x + 4 x + 4 x x x ln(x) 1 1089 INT ------------- dx = ---+4 --- +16 ---- +68 x - ----- + --- +---- ln(x-4) + C 3 2 4 3 2 4 x 4 x - 4 x ================================================== Fortunately .... The linear factors are easy to calculate. f(c) Simply use ------ . / g (c) 3x-1 A B ------------ = ----------- + -------- (x+2)(x-3) x+2 x-3 f(x) = 3x-1 g'(x) = x-3 + x+2 A = -7/(-5) = 7/5 B = 8/5 = 2x-1 ........................................................... 5x+3 A B C ------------- = ----------- + ---------- + -------- x(x-3)(x+1) x x-3 x+1 f(x) = 5x+3 g(x) = (x-3)(x+1) A = 3/(-3) = -1 +x (x+1) +x(x-3) B = 18/12 C = -2/4 ........................................................... 2 7x + 2 x - 3 A B C ------------------ = ------ + --------- + ---------- (2x-1)(3x+2)(x-3) x-1/2 x+2/3 x-3 / g (x) = 2(3x+2)(x-3) +(2x-1)3(x-3) +(2x-1)(3x+2) 7/4+1-3 -1/4 1 A = ------------- = ------ = ------ 2 7/2 (-5/2) -35/2 70 7(4/9)-4/3-3 -11/9 -1 B = -------------------- = ----- = ---- (-4/3-1)3(-2/3-3) 77/3 21 7/4+1-3 7(4/9)-4/3-3 63+6-3 -------- -------------- --------- 2 7/2 (-5/2) (-4/3-1)3(-2/3-3) 55 66 6 C = ----- = ---- 55 5 x A B -------- = --------- + ------------- 2 2 (x-3) (x-3) (x-3) It will not work on non linear roots. 2 6x -3x + 1 A Bx+C -------------------- = ---------- + -------------- 2 2 (4x+1)(x + 1) 4x+1 x + 1 2 6x -3x + 1 A B C -------------------- = ---------- + -------- + ------ 2 (4x+1)(x + 1) x+1/4 x + i x-i 2 f(x) = 6x -3x+1 / g (x) = 4(x^2+1) + (4x+1) 2x 6/16 +3/4 + 1 17/8 A = --------------- = ------ = 1/2 4(1/16+1) 17/4 -6+3i+1 -5+3i B = ------------- = ------ (-4i+1)(-2i) -8-2i -6-3i+1 -5-3i C = ------------ = ------ (4i+1)(2i) -8+2i -5+3i -5-3i 2 ------- ------- 6x -3x + 1 1/2 -8-2i -8+2i -------------------- = ---------- + -------- + ------ 2 (4x+1)(x + 1) x+1/4 x + i x-i 2 6x -3x + 1 1/2 x-1 -------------------- = ---------- + ---------- 2 2 (4x+1)(x + 1) x+1/4 x +1 --------------------------------------------------- Proof f(x) A B -------- = P(x) + ----- + ------- g(x) x-a x-b (x-a) f(x) B ------------- = (x-a) P(x) + A + (x-a) ---- g(x) x-b f(x) B ------------- = (x-a) P(x) + A + (x-a) ---- g(x)-g(a) x-b --------- x-a Taking the limit as x----->a f(a) ----- = A / g (a) Notice what happens when you have a double root. f(x) A B -------- = ----- + ------- g(x) x-a 2 (x-a) (x-a) f(x) B ---------- = A + -------- and the limit does not exist. g(x) (x-a) ---------------------------------------------------------- Page 412 Example 1 Set up the triangles for triangular substitution for these problems. _________ / 2 (a) INT \/ 9 - x dx __________ / 2 (b) INT \/ 16 - 4 y dx _________ / 4 (c) INT y \/ 1 - 4 y dy __________ t / 2t (d) INT e \/ 100 - e dt ----------------------------------------- Use this formula to integrate the previous problems: _______ ________ 2 _ _ / 2 2 u / 2 2 a | u | INT \/ a - u du = ------ \/ a - u + ---- ArcSin| --- | + C 2 2 |_ a _| On Page 420 Sample test problems: How would you approach each of the problems to do the integration.