2 3 3 2 sin2 sin2 1 1 2 2 cos xdx sin

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , G(0) = 0, so F (0) − G(0) = 8/3 (c) F (x) = G (x) = 10x/(x2 + 5)2 F (0) = 0, G(0) = −1, so F (0) − G(0) = 1 (c) 49. (a) (b) 48. F (x) = (9x2 + 24x + 16)/6 = 3x2 /2 + 4x + 8/3 = G(x) + 8/3 F (x) = x2 (x2 + 5) − 5 5 = =1− 2 = G(x) + 1 2+5 2+5 x x x +5 (sec2 x − 1)dx = tan x − x + C (csc2 x − 1)dx = − cot x − x + C 50. 51. (a) 1 2 52. (a) F (x) = G (x) = f (x), where f (x) = (b) G(x) − F (x) = (c) no, because (−∞, 0) ∪ (0, +∞) is not an interval 53. (1 − cos x)dx = 1087 v= √ 2 273 2, 3, 1 (x − sin x) + C 2 (b) 1 2 (1 + cos x) dx = 1 (x + sin x) + C 2 1, x > 0 −1, x < 0 x>0 so G(x) = F (x) plus a constant x<0 1087 1/2 1087 1/2 ft/s T −1/2 dT = √ T + C , v (273) = 1087 = 1087 + C so C = 0, v = √ T 273 273 213 Chapter 7 EXERCISE SET 7.3 1. u23 du = u24 /24 + C = (x2 + 1)24 /24 + C (a) (b) − u3 du = −u4 /4 + C = −(cos4 x)/4 + C (c) 2 √ sin u du = −2 cos u + C = −2 cos x + C 3 8 1 3 3 1/2 3 u +C = 4x2 + 5 + C 4 4 1 1 u−1 du = ln u + C = ln(x3 − 4) + C 3 3 (d) (e) 2. (a) (b) (c) 1 4 1 4 1 π (d) (e) 3. u−1/2 du = 1 1 tan u + C = tan(4x + 1) + C 4 4 1 3/2 1 u1/2 du = u + C = (1 + 2y 2 )3/2 + C 6 6 2 3/2 2 u1/2 du = u +C = sin3/2 πθ + C 3π 3π 5 5 u4/5 du = u9/5 + C = (x2 + 7x + 3)9/5 + C 9 9 du = ln u + C = ln(1 + ex ) + C u sec2 u du = 1 1 u du = − u2 + C = − cot2 x + C 2 2 1 10 1 (b) u9 du = u +C = (1 + sin t)10 + C 10 10 1 (c) du = ln |u| + C = ln | ln x| + C u 1 1 1 (d) − eu du = − eu + C = − e−5x + C 5 5 5 1 1 1 1 (e) − du = − ln |u| + C = − ln |(1 + cos 3θ)| + C 3 u 3 3 (a) − 2 7/2 4 5/2 2 3/2 u − u + u +C 7 5 3 4 2 2 = (1 + x)7/2 − (1 + x)5/2 + (1 + x)3/2 + C 7 5 3 (a) (u − 1)2 u1/2 du = (b) csc2 u du = − cot u + C = − cot(sin x) + C (c) 4. eu du = eu + C = etan x + C (d) (e) 5. 1 2 u1/2 du = (u5/2 − 2u3/2 + u1/2 )du = 1 3/2 1 u + C = (1 + e2t )3/2 + C 3 3 1 du = ln |u| + C = ln x5 + 1 + C u u = 2x, du = 2dx; 1 2 eu du = 1u 1 e + C = e2x + C 2 2 Exercise Set 7.3 214 1 2 1 1 1 du = ln |u| + C = ln |2x| + C u 2 2 6. u = 2x, du = 2dx; 7. u = 2 − x2 , du = −2x dx; − 8. u = 3x − 1, du = 3dx; 1 3 1 2 u3 du = −u4 /8 + C = −(2 − x2 )4 /8 + C u5 du = 16 1 u +C = (3x − 1)6 + C 18 18 9. u = 8x, du = 8dx; 1 8 cos u du = 10. u = 3x, du = 3dx; 1 3 1 1 sin u du = − cos u + C = − cos 3x + C 3 3 11. u = 4x, du = 4dx; 1 4 sec u tan u du = 12. u = 5x, du = 5dx; 1 5 sec2 u du = 13. u = 7t2 + 12, du = 14t dt; 1 14 u = x3 + 1, du = 3x2 dx; 1 3 16. u = 1 − 3x, du = −3dx; − 18. u = 3x2 , du = 6x dx; 1 6 1 10 1 3 2 1/2 2 u +C = 3 3 cos u du = 4 − 5x2 + C x3 + 1 + C 1 −1 1 u + C = (1 − 3x)−1 + C 3 3 u−3 du = − 1 −2 1 u + C = − (4x2 + 1)−2 + C 16 16 1 1 sin u + C = sin(3x2 ) + C 6 6 eu du = eu + C = esin x + C u = sin x, du = cos x dx; 20. u = x4 , du = 4x3 dx; 21. u = −2x3 , du = −6x2 , − 22. u = ex − e−x , du = (ex + e−x )dx, 23. u = 5/x, du = −(5/x2 )dx; − 24. u= 25. u = x3 , du = 3x2 dx; 1 4 eu du = 1 6 1 x, du = √ dx; 2 2x 1 3 1 3/2 1 u +C = (7t2 + 12)3/2 + C 21 21 1 1 u−1/2 du = − u1/2 + C = − 5 5 u−2 du = 19. √ 1 1 tan u + C = tan 5x + C 5 5 u−1/2 du = 1 8 17. u = 4x2 + 1, du = 8x dx; 1 1 sec u + C = sec 4x + C 4 4 u1/2 du = 14. u = 4 − 5x2 , du = −10x dx; − 15. 1 1 sin u + C = sin 8x + C 8 8 1u 14 e + C = ex + C 4 4 1 1 3 eu du = − eu + C = − e−2x + C 6 6 1 5 1 du = ln |u| + C = ln ex − e−x + C u sin u du = 1 1 cos u + C = cos(5/x) + C 5 5 √ sec2 u du = 2 tan u + C = 2 tan x + C sec2 u du = 1 1 tan u + C = tan(x3 ) + C 3 3 215 26. Chapter 7 u = cos 2t, du = −2 sin 2t dt; − 1 2 27. e−x dx; u = −x, du = −dx; − 28. ex/2 dx; u = x/2, du = dx/2; 2 1 3 1 1 u3 du = − u4 + C = − cos4 2t + C 8 8 eu du = −eu + C = −e−x + C √ eu du = 2eu + C = 2ex/2 + C = 2 ex + C u5 du = 16 1 u +C = sin6 3t + C 18 18 29. u = sin 3t, du = 3 cos 3t dt; 30. u = 5 + cos 2θ, du = −2 sin 2θ dθ; − 1 2 u−3 du = 31. u = 2 − sin 4θ, du = −4 cos 4θ dθ; − 1 4 1 1 u1/2 du = − u3/2 + C = − (2 − sin 4θ)3/2 + C 6 6 1 5 32. u = tan 5x, du = 5 sec2 5x dx; 33. u = sec 2x, du = 2 sec 2x tan 2x dx; 34. u = sin θ, du = cos θ dθ; 35. u= 36. u= 1 2 14 1 u +C = tan4 5x + C 20 20 13 1 u + C = sec3 2x + C 6 6 u2 du = sin u du = − cos u + C = − cos(sin θ) + C √ 1 y , du = √ dy , 2 2y eu du = 2eu + C = 2e √ 1 y , du = √ dy , 2 2y 1 du = 2 eu √ 38. u = a + bx, du = b dx, dx = 1 b u3 du = u1/n du = 1 −2 1 u + C = (5 + cos 2θ)−2 + C 4 4 y +C √ e−u du = −2e−u + C = −2e− y +C 1 du b n n u(n+1)/n + C = (a + bx)(n+1)/n + C b(n + 1) b(n + 1) 39. u = sin(a + bx), du = b cos(a + bx)dx 1 1 1 un du = un+1 + C = sinn+1 (a + bx) + C b b(n + 1) b(n + 1) 41. u = x − 3, x = u + 3, dx = du (u + 3)u1/2 du = (u3/2 + 3u1/2 )du = 2 5/2 2 u + 2u3/2 + C = (x − 3)5/2 + 2(x − 3)3/2 + C 5 5 42. u = y + 1, y = u − 1, dy = du 2 2 u−1 du = (u1/2 − u−1/2 )du = u3/2 − 2u1/2 + C = (y + 1)3/2 − 2(y + 1)1/2 + C 1/2 3 3 u 43. u = 3θ, du = 3 dθ 1 1 tan2 u du = 3 3 sin2 2θ sin 2θ dθ = 44. − 1 2 (se...
View Full Document

This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

Ask a homework question - tutors are online