Excavation costs 4 per m3 so the ttal cost of the

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Unformatted text preview: C = ln 2 +C +x 2 2 x +1 Ax + B Cx + D x3 + 3x2 + x + 9 =2 +2 ; A = 0, B = 3, C = 1, D = 0 so 2 + 1)(x2 + 3) (x x +1 x +3 1 x3 + 3x2 + x + 9 dx = 3 tan−1 x + ln(x2 + 3) + C (x2 + 1)(x2 + 3) 2 30. x3 + x2 + x + 2 Ax + B Cx + D =2 +2 ; A = D = 0, B = C = 1 so (x2 + 1)(x2 + 2) x +1 x +2 1 x3 + x2 + x + 2 dx = tan−1 x + ln(x2 + 2) + C (x2 + 1)(x2 + 2) 2 317 31. Chapter 9 x x3 − 3x2 + 2x − 3 =x−3+ 2 , 2+1 x x +1 1 1 x3 − 3x2 + 2x − 3 dx = x2 − 3x + ln(x2 + 1) + C 2+1 x 2 2 32. x x4 + 6x3 + 10x2 + x = x2 + 2 , 2 + 6x + 10 x x + 6x + 10 x dx = x2 + 6x + 10 = u−3 du, u2 + 1 x dx = (x + 3)2 + 1 u=x+3 1 ln(u2 + 1) − 3 tan−1 u + C1 2 1 1 x4 + 6x3 + 10x2 + x dx = x3 + ln(x2 + 6x + 10) − 3 tan−1 (x + 3) + C x2 + 6x + 10 3 2 so 1 A B 1 dx, and = + ; A = −1/6, x2 + 4x − 5 (x + 5)(x − 1) x+5 x−1 33. Let x = sin θ to get B = 1/6 so we get − 34. Let x = et ; then 1 1 dx + x+5 6 1 6 et dt = e2t − 4 1 x−1 1 1 dx = ln + C = ln x−1 6 x+5 6 1 − sin θ 5 + sin θ 1 dx, x2 − 4 A B 1 = + ; A = −1/4, B = 1/4 so (x + 2)(x − 2) x+2 x−2 − 1 x−2 1 et − 2 1 dx = ln + C = ln t + C. x−2 4 x+2 4 e +2 1 1 dx + x+2 4 1 4 2 35. V = π 0 18x2 − 81 x4 x4 =1+ 4 , dx, 4 2 )2 2 + 81 (9 − x x − 18x x − 18x2 + 81 B D 18x2 − 81 A C 18x − 81 + + = = + ; (9 − x2 )2 (x + 3)2 (x − 3)2 x + 3 (x + 3)2 x − 3 (x − 3)2 2 9 9 9 9 A = − , B = , C = , D = so 4 4 4 4 V =π x− 9/4 9 9/4 9 ln |x + 3| − + ln |x − 3| − 4 x+3 4 x−3 ln 5 36. Let u = ex to get − ln 5 dx = 1 + ex ln 5 − ln 5 A B 1 =+ ; A = 1, B = −1; u(1 + u) u 1+u 37. (x2 ex dx = x (1 + ex ) e 5 1/5 2 =π 0 5 1/5 19 9 − ln 5 5 4 du , u(1 + u) du = (ln u − ln(1 + u)) u(1 + u) 5 = ln 5 1/5 Cx + D Ax + B x2 + 1 +2 =2 ; A = 0, B = 1, C = D = −2 so 2 + 2x + 3) x + 2x + 3 (x + 2x + 3)2 (x2 x2 + 1 dx = + 2x + 3)2 1 dx − (x + 1)2 + 2 2x + 2 dx (x2 + 2x + 3)2 1 x+1 = √ tan−1 √ + 1/(x2 + 2x + 3) + C 2 2 + C. Exercise Set 9.5 38. 318 Cx + D x5 + x4 + 4x3 + 4x2 + 4x + 4 Ax + B Ex + F +2 =2 +2 ; 2 + 2)3 2 (x x +2 (x + 2) (x + 2)3 A = B = 1, C = D = E = F = 0 so √ 1 1 x+1 dx = ln(x2 + 2) + √ tan−1 (x/ 2) + C x2 + 2 2 2 39. x4 − 3x3 − 7x2 + 27x − 18 = (x − 1)(x − 2)(x − 3)(x + 3), A B C D 1 = + + + ; (x − 1)(x − 2)(x − 3)(x + 3) x−1 x−2 x−3 x+3 A = 1/8, B = −1/5, C = 1/12, D = −1/120 so 1 1 1 1 dx = ln |x − 1| − ln |x − 2| + ln |x − 3| − ln |x + 3| + C x4 − 3x3 − 7x2 + 27x − 18 8 5 12 120 40. 16x3 − 4x2 + 4x − 1 = (4x − 1)(4x2 + 1), A Bx + C 1 = +2 ; A = 4/5, B = −4/5, C = −1/5 so (4x − 1)(4x2 + 1) 4x − 1 4x + 1 16x3 1 1 1 dx = ln |4x − 1| − ln(4x2 + 1) − tan−1 (2x) + C 2 + 4x − 1 − 4x 5 10 10 41. (a) x4 + 1 = (x4 + 2x2 + 1) − 2x2 = (x2 + 1)2 − 2x2 √ √ = [(x2 + 1) + 2x][(x2 + 1) − 2x] √ √ √ √ = (x2 + 2x + 1)(x2 − 2x + 1); a = 2, b = − 2 (b) Ax + B Cx + D x √ √ √ = + ; 2x + 1)(x2 − 2x + 1) x2 + 2x + 1 x2 − 2x + 1 √ √ 2 2 , C = 0, D = so A = 0, B = − 4 4 √ √ 1 x 21 1 21 1 √ √ dx = − dx + dx 4+1 2+ 2− 4 0x 4 0x 2x + 1 2x + 1 0x √ √ 21 1 21 1 √ √ dx + dx =− 2 + 1/2 4 0 (x + 2/2) 4 0 (x − 2/2)2 + 1/2 √ √ √ √ 2 1+ 2/2 1 2 1− 2/2 1 du + du =− √ √ 2 + 1/2 2 + 1/2 4 u 4 − 2/2 u 2/2 (x2 + √ √ √ 1+ 2/2 1− 2/2 √ √ 1 1 + tan−1 2u √ = − tan−1 2u √ 2 2 2/2 − 2/2 √ √ 1 π 1 1π 1 + tan−1 ( 2 − 1) − − = − tan−1 ( 2 + 1) + 2 24 2 2 4 √ √ π1 = − [tan−1 ( 2 + 1) − tan−1 ( 2 − 1)] 4 2 √ √ π1 = − [tan−1 (1 + 2) + tan−1 (1 − 2)] 4 2 √ √ π1 (1 + 2) + (1 − 2) √ √ = − tan−1 (Exercise 46, Section 4.5) 4 2 1 − (1 + 2)(1 − 2) = π1 π1 − tan−1 1 = − 4 2 4 2 π 4 = π 8 319 42. Chapter 9 a2 1 2a B 1 1 1 A + ;A = ,B = so = 2 −x a−x a+x 2a 2a 1 1 + a−x a+x dx = 1 1 a+x (− ln |a − x| + ln |a + x| ) + C = ln +C 2a 2a a−x EXERCISE SET 9.6 1 2 + ln |2 − 3x| + C 9 2 − 3x 1. Formula (60): 3 4x + ln |−1 + 4x| + C 16 2. Formula (62): 3. Formula (65): x 1 ln +C 5 5 + 2x 4. Formula (66): − 5. Formula (102): 1 (x + 1)(−3 + 2x)3/2 + C 5 6. Formula (105): √ 1 4 − 3x − 2 +C 7. Formula (108): ln √ 2 4 − 3x + 2 x 2 x2 − 3 − √ 8. Formula (108): tan √ 12. Formula (93): − 13. Formula (95): x 2 14. Formula (90): − √ 16. Formula (80): − √ x2 + 5 + ln(x + x x2 + 4) + C 15. Formula (74): x 2 9 − x2 + 9 x sin−1 + C 2 3 4 − x2 x − sin−1 + C x 2 √ 20. Formula (40): − x−3 1 ln +C 6 x+3 x2 + 5) + C x2 − 2 +C 2x 3 − x2 − 18. Formula (117): − 3x − 4 +C 2 x2 − 3 + C x2 + 4 − 2 ln(x + √ 17. Formula (79): 10. Formula (70): 3 ln x + 2 √ 2 (−x − 4) 2 − x + C 3 −1 √ 1 x+ 5 √ +C 9. Formula (69): √ ln 25 x− 5 11. Formula (73): 1 1 − 5x − 5 ln +C x x √ √ 3 ln 3+ √ 9 − x2 x 6x − x2 +C 3x +C 19. Formula (38): − 1 1 sin(5x) + sin x + C 10 2 1 1 cos(7x) + cos(3x) + C 14 6 √1 22. Formula (50): 4 x ln x − 1 + C 2 21. Formula (50): x4 [4 ln x − 1] + C 16 23. Formula (42): e−2x (−2 sin(3x) − 3 cos(3x)) + C 13 Exercise Set 9.6 320 24. Formula (43): ex (cos(2x) + 2 sin(2x)) + C 5 4 1 u du = + ln 4 − 3e2x (4 − 3u)2 18 4 − 3e2x 1 2 25. u = e2x , du = 2e2x dx, Formula (62): 1 sin 2x du = ln +C 2u(3 − u) 6 3 − sin 2x 26. u = sin 2x, du = 2 cos 2xdx, Formula (116): √ 1 du −1 3 x = tan +C u2 + 4 3 2 √ 3 2 27. u = 3 x, du = √ dx, Formula (68): 3 2x 1 4 28. u = sin 4x, du = 4 cos 4xdx,...
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