Bài tập toán cao cấp tập 3 part 3 pps

66 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann Do d´o I = x · 1 cos x    π/3 0 − π/3  0 dx cos x = π 3 cos π 3 − ln tg  x 2 + π 4     π/3 0 = 2π 3 − ln tg  π 6 + π 4  + ln tg π 4 = 2π 3 −ln tg 5π 12 ·  V´ı du . 6. T´ınh t´ıch phˆan I = 1  0 x 2 (1 −x) 3 dx. Gia ’ i. Ta d ˘a . t u = x 2 ,dv=(1−x) 3 dx ⇒ du =2xdx, v = − (1 − x) 4 4 · Do d ´o I = −x 2 (1 − x) 4 4    1 0 + 1  0 2x (1 − x) 4 4 dx    I 1 =0+I 1 . T´ınh I 1 .T´ıch phˆan t`u . ng phˆa ` n I 1 ta c´o I 1 = 1 2 1  0 x(1 − x) 4 dx = − 1 2 x (1 − x) 5 5    1 0 + 1 2 1  0 (1 − x) 5 5 dx =0− 1 10 (1 − x) 6 6    1 0 = 1 60 ⇒ I = 1 60 ·  V´ı du . 7. ´ Ap du . ng cˆong th´u . c Newton-Leibnitz d ˆe ’ t´ınh t´ıch phˆan 1) I 1 = 100π  0 √ 1 − cos 2xdx, 2) I 2 = 1  0 e x arc sin(e −x )dx. 11.2. Phu . o . ng ph´ap t´ınh t´ıch phˆan x´ac d i . nh 67 Gia ’ i. Ta c´o √ 1 − cos 2x = √ 2|sinx|.Dod´o 100π  0 √ 1 − cos 2xdx = √ 2 100π  0 |sinx|dx = √ 2  π  0 sin xdx − 2π  π sin xdx + 3π  2π sin xdx − + ···+ 100π  99π sin xdx  = − √ 2[2 + 2 + ···+ 2] = 200 √ 2. 2) Thu . . chiˆe . n ph´ep d ˆo ’ ibiˆe ´ n t = e −x , sau d´o ´a p d u . ng phu . o . ng ph´ap t´ıch phˆan t`u . ng phˆa ` n. Ta c´o  e x arc sin(e −x )dx = −  arc sin t t 2 dt = 1 t arc sin t −  dt t √ 1 − t 2 = 1 t arc sin t + I 1 . I 1 = −  dt t √ 1 − t 2 =  d  1 t    1 t  2 − 1 =ln  1 t +  1 t 2 −1  + C. Do d ´o  e x arc sin e −x dx = arc sin t t +ln  1 t +   1 t 2 − 1  + C = e x arc sin e −x + ln(e x + √ e 2x − 1) + C Nguyˆen h`am v`u . athud u . o . . c c´o gi´o . iha . nh˜u . uha . nta . id iˆe ’ m x = 0. do d ´o theo cˆong th´u . c (11.3) ta c´o 1  0 e x arc sin e −x dx = earc sin e −1 − π 2 + ln(e + √ e 2 − 1).  68 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann V´ı du . 8. T´ınh t´ıch phˆan Dirichlet π/2  0 sin(2n − 1)x sin x dx, n ∈ N. Gia ’ i. Ta c´o cˆong th´u . c 1 2 + n−1  k=1 cos 2kx = sin(2n −1)x 2 sin x · T`u . d ´o v`a lu . u´yr˘a ` ng π/2  0 cos 2kxdx =0,k =1, 2, ,n− 1 ta c´o π/2  0 sin(2n − 1)x sin x dx = π 2 ·  B ` AI T ˆ A . P T´ınh c´ac t´ıch phˆan sau d ˆay b˘a ` ng phu . o . ng ph´ap d ˆo ’ ibiˆe ´ n (1-14). 1. 5  0 xdx √ 1+3x .(D S. 4) 2. ln 3  ln 2 dx e x − e −x .(DS. ln 3 2 2 ) 3. √ 3  1 (x 3 +1)dx x 2 √ 4 − x 2 .(DS. 7 2 √ 3 − 1). D ˘a . t x = 2 sin t. 4. π/2  0 dx 2 + cos x .(D S. π 3 √ 3 ) 11.2. Phu . o . ng ph´ap t´ınh t´ıch phˆan x´ac d i . nh 69 5. 1  0 x 2 dx (x +1) 4 .(DS. 1 24 ) 6. ln 2  0 √ e x − 1dx.(DS. 4 − π 2 ) 7. √ 7  √ 3 x 3 dx 3  (x 2 +1) 2 .(DS. 3) Chı ’ dˆa ˜ n. D ˘a . t t = x 2 +1. 8. e  1 4 √ 1+lnx x dx.(D S. 0, 8(2 4 √ 2 − 1)) Chı ’ dˆa ˜ n. D ˘a . t t =1+lnx. 9. + √ 3  −3 x 2 √ 9 − x 2 dx.(DS. 81π 8 ) chı ’ dˆa ˜ n. D ˘a . t x = 3 cos t. 10. 3  0  x 6 − x dx.(D S. 3(π − 2) 2 ) Chı ’ dˆa ˜ n. D ˘a . t x = 6 sin 2 t. 11. π  0 sin 6 x 2 dx.(D S. 5π 16 ) Chı ’ dˆa ˜ n. D ˘a . t x =2t. 12. π/4  0 cos 7 2xdx.(DS. 8 35 ) Chı ’ dˆa ˜ n. D ˘a . t x = t 2 70 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann 13. √ 2/2  0  1+x 1 − x dx.(D S. π 4 +1− √ 2 2 ) Chı ’ dˆa ˜ n. D ˘a . t x = cos t. 14. 29  3 3  (x − 2) 2 3+ 3  (x − 2) 2 dx.(DS. 8 + 3 √ 3 2 π) T´ınh c´ac t´ıch phˆan sau d ˆay b˘a ` ng phu . o . ng ph´ap t´ıch phˆan t`u . ng phˆa ` n (15-32). 15. 1  0 x 3 arctgxdx.(DS. 1 6 ) 16. e  1/e |ln x|dx.(DS. 2(1 −1/e)) 17. π  0 e x sin xdx.(DS. 1 2 (e π + 1)) 18. 1  0 x 3 e 2x dx.(DS. e 2 +3 8 ) 19. 1  0 arc sin x √ 1+x dx.(D S. π √ 2 − 4) 20. π/4  0 ln(1 + tgx)dx.(DS. π ln 2 8 ) 21. π/b  0 e ax sin bxdx.(DS. b a 2 + b 2  e πa b +1  ) 22. 1  0 e −x ln(e x +1)dx.(DS. − 1+e e ln(e +1)+2ln2+1) 11.2. Phu . o . ng ph´ap t´ınh t´ıch phˆan x´ac d i . nh 71 23. π/2  0 sin 2x · arctg(sin x)dx.(DS. π 2 − 1) 24. 2  1 sin(ln x)dx.(DS. sin(ln 2) −cos(ln 2) + 1 2 ) 25. π  0 x 3 sin xdx.(DS. π 3 − 6π) 26. 2  1 xlog 2 xdx.(DS. 2 − 3 4ln2 ) 27. a √ 7  0 x 3 3 √ a 2 + x 2 dx.(DS. 141a 3 3 √ a 20 ) 28. a  0 √ a 2 −x 2 dx.(DS. πa 2 4 ) 29. π/2  π/6 x + sin x 1 + cos x dx.(D S. π 6 (1 + √ 3)) 30. π/2  0 sin m x cos(m +2)xdx.(DS. − cos mπ 2 m +1 ) 31. π/2  0 cos m x cos(m +2)xdx.(DS. 0) 32. π/2  0 cos x cos 2nxdx.(DS. π 4n (−1) n−1 ) 72 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann 33. T´ınh 2  0 f(x)dx, trong d´o f(x)=    x 2 khi 0  x  1 2 − x khi 1  x  2 b˘a ` ng hai phu . o . ng ph´ap; a) su . ’ du . ng nguyˆen h`am cu ’ a f(x) trˆen d oa . n [0, 2]; b) chia d oa . n[0, 2] th`anh hai doa . n[0, 1] v`a [1, 2]. (DS. 5 6 ) 34. Ch ´u . ng minh r˘a ` ng nˆe ´ u f(x)liˆen tu . ctrˆend oa . n[−, ]th`ı (i)   − f(x)dx =2   0 f(x)dx khi f(x) l`a h`am ch˘a ˜ n; (ii)   − f(x)dx = 0 khi f(x) l`a h`am le ’ . 35. Ch´u . ng minh r˘a ` ng ∀m, n ∈ Z c´ac d ˘a ’ ng th´u . c sau d ˆay du . o . . c tho ’ a m˜an: (i) π  −π sin mx cos nxdx =0. (ii) π  −π cos mx cos nxdx =0,m = n. (iii) π  −π sin mx sin nxdx =0,m = n. 36. Ch ´u . ng minh d ˘a ’ ng th´u . c b  a f(x)dx = b  a f(a + b − x)dx. Chı ’ dˆa ˜ n. D ˘a . t x = a + b − t. 11.2. Phu . o . ng ph´ap t´ınh t´ıch phˆan x´ac d i . nh 73 37. Ch´u . ng minh d ˘a ’ ng th´u . c π/2  0 f(cos x)dx = π/2  0 f(sin x)dx. Chı ’ dˆa ˜ n. D ˘a . t t = π 2 − x. 38. Ch´u . ng minh r˘a ` ng nˆe ´ u f(x)liˆen tu . c khi x  0th`ı a  0 x 3 f(x 2 )dx = 1 2 a 2  0 xf(x)dx. 39. Ch´u . ng minh r˘a ` ng nˆe ´ u f(t) l`a h`am le ’ th`ı x  a f(t)dt l`a h`am ch˘a ˜ n, t´u . cl`a −x  a f(t)dt = x  a f(t)dt. Chı ’ dˆa ˜ n. D ˘a . t t = −x v`a biˆe ’ udiˆe ˜ n −x  −a f(t)dt = a  −a + −x  a v`a su . ’ du . ng t´ınh ch˘a ˜ nle ’ cu ’ a h`am f. T´ınh c´ac t´ıch phˆan sau d ˆay (40-65) b˘a ` ng c´ach ´ap du . ng cˆong th´u . c Newton-Leibnitz. 40. 5  0 xdx √ 1+3x .(D S. 4) 41. ln 3  ln 2 dx e x − e −x .(DS. ln 1, 5 2 ) 74 Chu . o . ng 11. T´ıch phˆan x´ac d i . nh Riemann 42. √ 3  0 (x 3 +1)dx x 2 √ 4 − x 2 .(DS. 7 2 √ 3 − 1) 43. π/2  0 dx 2 + cos x .(D S. π 3 √ 3 ) 44. ln 2  0 √ e x − 1dx.(DS. 4 − π 2 ) 45. √ 7  √ 3 x 3 dx 3  (x 2 +1) 2 .(DS. 3) 46. e  1 4 √ 1+lnx x dx.(D S. 0, 8(2 4 √ 2 − 1)) 47. 3  −3 x 2 √ 9 − x 2 dx.(DS. 81π 8 ) 48. 3  0  x 6 − x dx.(D S. 3(π − 2) 2 ) Chı ’ dˆa ˜ n. D ˘a . t x = 6 sin 2 t. 49. 4  3 x 2 +3 x − 2 dx.(D S. 11 2 + 7ln2) 50. −1  −2 x +1 x 2 (x − 1) dx.(D S. 2 ln 4 3 − 1 2 ) 51. 1  0 (x 2 +3x)dx (x + 1)(x 2 +1) .(D S. π 4 ) 11.2. Phu . o . ng ph´ap t´ınh t´ıch phˆan x´ac d i . nh 75 52. 1  0 dx √ x 2 +2x +2 .(D S. ln 2+ √ 5 1+ √ 2 ) 53. 4  0 dx 1+ √ 2x +1 .(D S. 2 − ln 2) 54. 2  1 e 1 x x 3 dx.(DS. 1 2 (e − e 1 4 )) 55. e  1 dx x(1 + ln 2 x) .(D S. π 4 ) 56. e  1 cos(ln x) x dx.(D S. sin 1) 57. 1  0 xe −x dx.(DS. 1 − 2 e ) 58. π/3  π/4 xdx sin 2 x .(D S. π(9 − 4 √ 3) 36 ) 59. 3  1 ln xdx.(DS. 3 ln 3 − 2) 60. 2  1 x ln xdx.(DS. 2 ln 2 − 3 4 ) 61. 1/2  0 arc sin xdx.(DS. π 12 + √ 3 2 −1) 62. π  0 x 3 sin xdx.(DS. π 3 − 6π) [...]... t ∈ − , Tu.o.ng tu nhu trˆn ta ´p dung (11. 23) ´ o e a 2 2 o.c v` thu du a π/2 Sy = 2π 6 cos t · 36 sin2 t + 9 cos2 tdt 1 D˘t sin t = √ shϕ a 3 −π/2 √ = 24 3 √ arcsh 3 √ √ √ ch2ϕdϕ = 24 3 2 3 + ln(2 + 3) √ −arcsh 3 ` ˆ BAI TAP ’ o T´ dˆ d`i cung cua du.`.ng cong ınh o a √ 8 ´ (10 10 − 1)) u e 1 y = x3/2 t` x = 0 dˆn x = 4 (DS 27 ’ ıch a a i 11 .3 Mˆt sˆ u.ng dung cua t´ phˆn x´c d nh o o´... elip: ´ ´ Dˆ a o e o a o u e o o x = 6 cos t, y = 3 sin t, 0 t 2π e ’ e e ’ 1+ Ph´p quay xung quanh truc Ox Ta x´t nu.a trˆn cua elip tu.o.ng ´ u.ng v´.i 0 t π Theo cˆng th´.c (11.22) du.´.i dang tham sˆ ta c´ ´ o o u o o o π Sx = 2π 3 sin t · 36 sin2 t + 9 cos2 tdt 0 2 D˘t cos t = √ sin ϕ ta c´ o a 3 √ Sx = 24 3 π /3 √ √ cos2 ϕdϕ = 2 3 (4π + 3 3) −π /3 ’ ’ e 2+ Ph´p quay xung quanh truc Oy Ta x´t nu.a... 2 Do d´ o a 0 y 2dx = −6a3π V = 2π 0 sin6 t cos2 t sin tdt π/2 0 = 6a3π (1 − cos2 t )3 cos2 t(− sin tdt) π/2 0 3 (cos2 t − 3 cos4 t + 3 cos6 t − cos8 t)(d(cos t) = 6a π π/2 = ··· = 32 3 πa 105 ’ ’ ’ o ` V´ du 6 T´ thˆ t´ vˆt thˆ gi´.i han bo.i hypecboloid mˆt tˆng ı ınh e ıch a e o a x2 y 2 z 2 + 2 − 2 =1 a2 b c ’ ıch a a i 11 .3 Mˆt sˆ u.ng dung cua t´ phˆn x´c d nh o o´ ´ ’ v` c´c m˘t ph˘ng z... cho tru.´.c xung quanh truc cho tru.´.c o o o 23 D : y 2 = 2px, x = a; xung quanh truc Ox (DS πpa2) ’ ıch a a i 11 .3 Mˆt sˆ u.ng dung cua t´ phˆn x´c d nh o o´ ´ x2 y 2 24 D : 2 + 2 a b x2 y 2 25 D : 2 + 2 a b 26 27 28 29 30 31 32 4π 2 a b) 1 (b < a) xung quanh truc Oy (DS 3 4π 2 ab ) 1 (b < a) xung quanh truc Ox (DS 3 2 D : 2y = x2 ; 2x + 2y − 3 = 0 xung quanh truc Ox (DS 18 π) 15 π 2 2 D :... (DS e u 2 2e π 1 ´ y = ln cos x t` x = 0 dˆn x = (DS ln 3) u e 6 2 π 2π ´ (DS ln 3) y = ln sin x t` x = dˆn x = u e 3 3 √ π (DS 2(eπ/2 − 1)) x = et sin t, y = et cos t, 0 t 2 x = a(t − sin t), y = a(1 − cos t); 0 t 2π (DS 8a) ´ 2 y = x2 − 1 t` x = −1 dˆn x = 1 (DS u e 3 4 5 6 7 8 x = a cos3 t, y = a sin3 t; 0 ˜ ’ a Chı dˆ n V` ı t 2π (DS 6a) 3a a a o y xt 2 + yt2 = | sin 2t| v` h`m | sin 2t| c´ chu... du.`.ng xycloid ınh o ’ o a a 3 a2 ) 8 14 x = a cos t, y = b sin t, t ∈ [0, 2π] (DS πab) 13 x = a cos3 t, y = a sin3 t, t ∈ [0, 2π] (DS o 15 Du.`.ng lemniscate Bernoulli ρ2 = a2 cos 2ϕ (DS a2 ) o ınh 16 Du.`.ng h` tim (Cacdioid) ρ = a(1 + cos ϕ) 3 a2 ) (DS 2 87 ıch a a Chu.o.ng 11 T´ phˆn x´c dinh Riemann 88 √ o o 17∗ C´c du.`.ng tr`n ρ = 2 3a cos ϕ, ρ = 2a sin ϕ a √ 5 (DS a2 π − 3 ) 6 ’ ’ Trong c´c b`i... du.`.ng astroid ı ım e ıch ınh a o 3 x = a cos3 t, y = a sin t ´ ’ ´ Giai Ap dung cˆng th´.c (11.7) V` du.`.ng astroid dˆi x´.ng qua o u ı o o u ıch a a Chu.o.ng 11 T´ phˆn x´c dinh Riemann 82 c´c truc toa dˆ (h˜y v˜ h`nh !) nˆn a e o a e ı 0 a sin3 t · 3a cos2 t(− sin t)dt S = 4S1 = 4 π/2 π/2 = 12a2 sin4 t cos2 tdt 0 π/2 3 = a2 2 (1 − cos 2t)(1 − cos2 2t)dt 0 3 = 3 a 8 ’ V´ du 2 Trˆn hypecbon x2... dy b2 0 y3 = 2πa2 y + 2 3b b 0 8 = πa2b 3 ’ ’ V´ du 5 T´ thˆ t´ vˆt thˆ lˆp nˆn do quay astroid x = a cos3 t, ı ınh e ıch a e a e y = a sin3 t, 0 t 2π xung quanh truc Ox ´ ´ ’ Giai Du.`.ng astroid dˆi x´.ng dˆi v´.i c´c truc Ox v` Oy Do d´ o a o u o o a o a a 2 Vx = π y 2 dx y dx = 2π −a 0 y 2 = a2 sin6 t, dx = −3a cos2 t sin tdt π t = khi x = 0, t = 0 khi x = a 2 Do d´ o a 0 y 2dx = −6a3π V = 2π... π dx = √ 4 + x2 + 1 x 2 2 0 pi/2 dx =1 1 + cos x 71 0 1 72 0 √ √ 1 1 x2 + 1dx = √ + ln(1 + 2) 2 2 a ınh ıch a a i 11.2 Phu.o.ng ph´p t´ t´ phˆn x´c d nh 1 73 1− √ 3 x2 3/ 2 dx = 3 32 0 D˘t x = sin3 ϕ a a x2 74 a−x π 2 2 dx = − a , a > 0 a+x 4 3 0 D˘t x = a cos ϕ a 2a √ πa2 2 dx = 2ax − x 2 75 0 D˘t x = 2a sin2 ϕ a 1 π ln(1 + x) dx = ln 2 2 1+x 8 76 0 ˜ ` a ’ a Chı dˆ n D˘t x = tgt rˆi ´p dung cˆng... (3 − 4) · 6 ’ ıch a a i 11 .3 Mˆt sˆ u.ng dung cua t´ phˆn x´c d nh o o´ ´ ` ˆ BAI TAP ’ a ınh e ı a ı a Trong c´c b`i to´n sau dˆy (1-17) t´ diˆn t´ch c´c h`nh ph˘ng a a a ’ a gi´.i han bo.i c´c du.`.ng d˜ chı ra o o a ’ 9 (DS ) 1 y = 6x − x2 − 7, y = x − 3 2 2 2 y = 6x − x , y = 0 (DS 36 ) 5 (DS 5 ) 3 4y = 8x − x2 , 4y = x + 6 24 2 2 4 y = 4 − x , y = x − 2x (DS 9) 5 6x = y 3 − 16y, 24x = y 3 . (1-14). 1. 5  0 xdx √ 1+3x .(D S. 4) 2. ln 3  ln 2 dx e x − e −x .(DS. ln 3 2 2 ) 3. √ 3  1 (x 3 +1)dx x 2 √ 4 − x 2 .(DS. 7 2 √ 3 − 1). D ˘a . t x = 2 sin t. 4. π/2  0 dx 2 + cos x .(D S. π 3 √ 3 ) 11.2 x .(D S. π 3 √ 3 ) 44. ln 2  0 √ e x − 1dx.(DS. 4 − π 2 ) 45. √ 7  √ 3 x 3 dx 3  (x 2 +1) 2 .(DS. 3) 46. e  1 4 √ 1+lnx x dx.(D S. 0, 8(2 4 √ 2 − 1)) 47. 3  3 x 2 √ 9 − x 2 dx.(DS. 81π 8 ) 48. 3  0  x 6. − 2 e ) 58. π /3  π/4 xdx sin 2 x .(D S. π(9 − 4 √ 3) 36 ) 59. 3  1 ln xdx.(DS. 3 ln 3 − 2) 60. 2  1 x ln xdx.(DS. 2 ln 2 − 3 4 ) 61. 1/2  0 arc sin xdx.(DS. π 12 + √ 3 2 −1) 62. π  0 x 3 sin xdx.(DS.