Tài liệu Bài tập về nguyên hàm tích phân

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Tham gia: 10/08/2016

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I. TÍNH TÍCH PHÂN BẰNG CÁCH SỬ DỤNG TÍNH CHẤT VÀ NGUYÊN HÀM CƠ BẢN: 1 e 3 ( x  x  1)dx 3 3. 1 (2sin x  3cosx  x)dx  3 8.  5. 7. 0 1 (3sin x  2cosx  x )dx  3 (e 2 3 ( x  x x  x )dx 1 ( 12. 1 9. (e 14. x 2 11. ( 13. 2 7x  2 x  5 dx 1 x  x 2  1)dx 0  1).dx 1 e x  1)( x  x  1)dx x.dx 2 2 x -1 5 15. x  1)( x  x  1)dx 1 2 3  x) dx 1 3 (x x 0 2 10. x  1dx 1 2 3 ( x  x x )dx  2  x 2 )dx 1 1 6. 2 2  x  2 dx  2 4. 1 2. 1 1. 0 2. 1 ( x  x  x  2 dx x2  x2  2 2 16. ( x  1).dx 1 x 2  x ln x  4 18.  0 tgx .dx cos2 x 1 20.  0 ln 3 22.  0 17. .dx ex  e x II. PHƯƠNG PHÁP ĐẶT ẨN PHỤ: 6 1 19. e x  e x 0 ex  e x dx 2 e x .dx e x  e x cos3 x.dx  3 sin x 21.  1  2 22. dx 4x 2  8x dx 0 1  sin x  2 1. sin  3  2 xcos 2 xdx 2. 3  2 3. sin x 0 1  3cosx dx 3. 5. 1 6. tgxdx 0  x x 2  1dx 0 3 2 x x  1dx 0 0 7. 9. x 1  x 2 dx 1 x2 0  3 2 x 1  x dx 0 2 11. 1 12. 1 0 1  x 2 dx 1 14.  2 16. x2  1 0 e  sin x 18. 20. 22. cosxdx 2 xdx 19. sin x x e cosxdx 21. 2 2 xdx 23.  2 e  25. dx cosx sin xdx 4 sin  3 xcos 2 xdx 3 e  cosx sin xdx 4 sin  3 xcos 2 xdx 3  2 2 3 sin xcos xdx 24. 2 2  2 0  3 1 0  2 4 1 15. dx 1 dx  2x  2 (1  3x )  2 2 2 1  2 0 e  13. 17. x e  2 1 dx 4 1 x3  1 1 x dx 1 x 1 1  x3  1 0 1 10. 1  4sin xcosxdx 1 1 8. xcos 3 xdx 3  6 cot gxdx  6 2  4  4 4. sin  sin x 1  3cosx dx 0  4  4 26. 27. 0  6 28. cot gxdx tgxdx  1 1  4sin xcosxdx 29. 0 1 30. 2 x 1  x dx 0 1 32.  0 2 34. x 1 x2 x 1 3 31. 1 x3  1 33. dx 35. e2ln x 1 dx  38. 1 x 37. 1 e cos2 (1  ln x) dx 42. 0 1 44.  0 3 46.  1 x 3 x 2  1dx x 3 1  x 2 dx e 1  ln x dx x 0 x dx 2x 1 1 dx x 1  x x 1 dx x 39. e2ln x 1 dx  49. 1 x  1 1  3ln x ln x dx x  1 1  ln 2 x e x ln x dx 2 41. 1  1 x dx x 1 1 43. x 45. 1 dx x 1  x  0 e  46. 1 e 48. x  1dx 0 1 e sin(ln x) 1 x dx 47. 0 e2 2  0 e e 1 x 2  1dx 1 dx sin(ln x) 1 x dx 36. 40. x 1 e e  6 1  ln x dx x 1  3ln x ln x dx x  1 e2 e 50. 1  ln 2 x e x ln x dx 1 e 51. 1 e cos 2 (1  ln x) dx  x 2 x 3  5dx 2 52. 0  2 53.  sin 4 4  54. x  1 cos xdx 0 0 4  55. 4  x 2 dx 1 dx 2 1  x 0  56. 4  x dx 2 0 II. PHƯƠNG PHÁP TÍCH PHÂN TỪNG PHẦN: b b u( x)v'(x)dx  u( x)v( x) a  v( x)u '( x)dx b Công thức tích phân từng phần : a Tích phân các hàm số dễ phát hiện u và dv a sin ax     f ( x) cosax dx e ax   @ Dạng 1  u  f ( x)  sin ax        dv  cos ax  dx  e ax    du  f '( x)dx  sin ax       v  cosax  dx  eax    @ Dạng 2:  f ( x) ln(ax)dx  dx   u  ln( ax)  du  x     dv  f ( x)dx  v  f ( x)dx   Đặt  ax sin ax  e  . cosax dx  @ Dạng 3: Ví dụ 1: tính các tích phân sau u  x2e x  1 x 2e x dx  0 ( x  1)2 dx  dv  ( x  1)2 a/ đặt 5 u  x   x 3dx  dv  ( x 4  1)3  3 x8 dx  4 3 b/ 2 ( x  1) đặt 1 c/ 1 1 1 dx 1  x2  x2 dx x 2 dx  dx   0 (1  x2 )2 0 (1  x2 )2 0 1  x2 0 (1  x2 )2  I1  I 2 1 Tính I1 dx  2 0 1 x bằng phương pháp đổi biến số u  x  x dx x  0 (1  x 2 )2  dv  (1  x 2 )2 dx Tính I2 = bằng phương pháp từng phần : đặt  1 2 Bài tập e e 3 ln x dx 3 x 1. 1  2. 1 3. x ln( x 2 e 4. 6. 1 7. 2 x ln( x  1)dx 0  2 9. ( x  cosx) s inxdx 0 x 2 ln xdx 1 e 3 ln x 1 x3 dx 5. 1 e  1)dx 0 x ln xdx x ln xdx 1 e 8. x 2 ln xdx 1 e 1 ( x  1 x ) ln xdx 10.  3 x tan  2 ln( x  x)dx 2 11. 1 2 ln x dx x5  13. 1 12.  III. TÍCH PHÂN HÀM HỮU TỶ: 4 x cos xdx  14. 0  2 xe x dx 0 xdx  2 1 15. 2  e x cos xdx 16. 0 5 1. 2x  1 3 x 2  3x  2 dx b 2. 1 3. x3  x 1 0 x  1 dx 1 5. 2 1  x 2008 1 x(1  x 2008 ) dx 7. 3 x4 2 ( x 2  1) 2 dx 9. 2 x2  3 1 x( x 4  3x 2  2) dx 11. 2 1 0 4  x 2 dx 13. 2 1 0 x 2  2 x  2dx 15. 4 1 2 x 3  2 x 2  x dx 17. 2 1 x2 1 1  x 4 dx 19. 1 x6  x5  x4  2 dx 0 x6 1 21. a 1 4. x3  x 1 0 x 2  1 dx 1 2 x 0 (3x  1) 3 dx 1  ( x  a)( x  b) dx 6.  ( x  2) 0 0 2x 3  6x 2  9x  9 1 x 2  3x  2 dx  8. 1 x 2 n 3 0 (1  x 2 ) n dx 10. 2 1 1 x(1  x 4 ) dx 12. 1 14. 16. 1 25. 27. 29. 31. 33. 35. 37. dx x2  x  1  0 4 dx 0 1 x  (1  x 0 2 3 ) dx 3 3x 2  3x  3 2 x 3  3x  2 dx 18. 1 1 0 1  x 3 dx 20. 1 2  x4 0 1  x 2 dx 22. 1 23. x 1 x 1 1 x4 0 1  x 6 dx 1 dx ( x  3) 2 2 24. 26. 28. 30. 32. 34. 36. 38. 4 x  11 dx x2  5x  6  0 39. 40. IV. TÍCH PHÂN HÀM LƯỢNG GIÁC:  2 1.  sin x cos 4 xdx 2. 0  2 3. 2  sin 4 4. 0  cos 2 x(sin 4 6.  2 8. 3  (sin  (2 sin  (sin 10 x  cos 10 x  cos 4 x sin 4 x)dx 1  2  sin x dx 0  3  2 11. x  sin x cos x  cos 2 x)dx 0 10. sin 3 x 0 1  cos 2 x dx 2 0  2 dx 0 2  cos x x  cos 3 )dx 3 0  2 9. x cos 3 xdx 0  2 1  sin x dx 7. 2  2 x  cos 4 x)dx 0  sin  2 x cos 5 xdx  2 5.  2   sin 12. 4 dx x. cos x 6  4 13.  2 15. cos x  2  cos x dx 0 cos 3 x 0 1  cos x dx  2 16. 19. 18. 3 21. 0 0 sin x 0 1  sin x  cos x  1 dx 0  2 sin x  cos x  1  sin x  2 cos x  3 dx 2 20.   4  4 3  tg xdx cos x  1  cos x dx  2  sin x dx  2 cos xdx   (1  cos x ) 14.  2  2 17.  2 dx 0 sin 2 x  2 sin x cos x  cos 2 x 2  cot g  22. 6 3 xdx  3  4 4  tg xdx 23.  4 24.  4  0 25. 2 27.   ) 4 sin x  7 cos x  6  4 sin x  5 cos x  5 dx 0  4 28. dx  2 sin x  3 cos x  4 sin x dx 4 x  2 1  cos 2 x  sin 2 x dx sin x  cos x 0  30.  2  2 31. sin 3x 0 1  cos x dx sin 3 x 0 cos 2 x dx 32. 4  2  sin 2 x(1  sin 34. 35.  37. sin x dx  4 36. 0  2 0 38. 0  4 3 5  cos x sin xdx 39.   0  6 dx 0 5 sin x  3  3 sin 4 xdx 2 x  1  cos 40.  2 41. dx  2 sin x  1  2  4   sin 2.  sin x sin( x  ) 6 43. 6   4. 4 dx x cos x 6  3 dx x) 3 dx sin 3 x  sin x dx sin 3 xtgx  2 dx  1  sin x  cos x 2 0  3 3   cos x dx   sin 2 x  sin x  4 33. 13 0 3  1  cos 0 26. 1  sin x dx 0  4 29. cos x cos( x  0  2 dx 1  1  tgx dx 4 dx sin x cos( x   ) 4  3  3 sin 2 xdx  6  cos x 45. 46. 4  3 47.  sin 48.  3 x dx 50. 0  sin 2 x.e 2 x 1 x 52. 0 54. 2 55. 57. 56. 1  2  (2 x  1) cos 2 xdx 58. 0  sin 0 61. e s in2 x 62. 2 0 64.  2   2 6 2 sin 2 xdx x  5 sin x  6 ln(sin x) dx cos 2 x  x sin x cos 2 xdx 0 e 2x sin 2 xdx 0  ln(1  tgx )dx 0 (1  sin x ) cos x  (1  sin x)(2  cos 0 2 x) dx  2 sin 2x sin 7 xdx 65.    2 dx  (sin x  2 cos x) dx  4 sin x cos 3 xdx 0  4 63. 60. 0  2 3  2  xtg xdx x 0   4 59. 1  sin x  1  cos x e   cos(ln x)dx cos xdx  2 sin 3x sin 4 x  tgx  cot g 2 x dx 6 2 0  2 dx 2 2  2  4 53. sin 2 x  (2  sin x) 4 sin xdx 0 (sin x  cos x) 3  2 51. 6 0  2 49.   tgxtg ( x  6 )dx   cos x(sin 4 x  cos 4 x) dx 66. 0  2 4sin 3 x dx 1  cos x 0  67. 69. 71. 73. 75. 77. 79. 68. 70. 72. 74. 76. 78. 80. V. TÍCH PHÂN HÀM VÔ TỶ: b  R( x, f ( x)) dx Trong ®ã R(x, f(x)) cã c¸c d¹ng: a ax   [0; ] a  x ) §Æt x = a cos2t, t 2 a sin t a cos t a2  x2 +) R(x, +) R(x, ) §Æt x = n +) R(x, ax  b cx  d ) §Æt t = n hoÆc x = ax  b cx  d 1 2 2 +) R(x, f(x)) = (ax  b) x  x   Víi ( x  x   )’ = k(ax+b) 1 2  x   x   Khi ®ã ®Æt t = , hoÆc ®Æt t = ax  b    [ ; ] 2 2 a tgt 2 2 +) R(x, a  x ) §Æt x = ,t   [0;  ] \ { } x  a ) §Æt x = cos x , t 2 a +) R(x,  +) R n1 2 2 n n x ; 2 x ;...; i x §Æt x = tk  Gäi k = BCNH(n ; n ; ...; n ) 1 2 2 3 1.  5 1 2 3. x x2  4  (2 x  3)  1 2  dx 2. dx 4 x  12 x  5 2 4. x x2 1 3 2 2 dx x 1 dx x3  1 2 i 2 2 5.  x 2  2008dx 1 6. 1 7. 2 2  x 1  x dx 0 3 9. x 1 x 2  2008 1 1 8.  (1  x 2 ) 3 dx 0 2 2 x 1 2 2 1 11. dx  x2 1 10. 0 2 2 dx  (1  x 2 ) 3 0  12. 2 2 15. 7  cos 2 x 0  2 17. 2  cos 2 x 0 7 19.  3 0  25. 20. 0  sin 2 x  sin x 0 x  10  x 2 dx 3 x 3 dx  2 22. 0 x  x  1 1 24. x  26. 2 27. 11  x  x  1 0 ln 2 dx 15 1  3 x 8 dx 0 1  cos 3 x sin x cos 5 xdx 0 1 dx 0 ln 3 6 1  3 cos x 1 2x  1  1 2  2 dx  cos x  cos 2 x dx 3 2x  1 0  2 18. xdx 7 23. 1 x  sin x  2 x 3 dx 1 21. 16. cos xdx  1 x2 0  2 cos xdx  x 2 dx  14.  2 (1  x 2 ) 3 0 1  x 2 dx 0 dx  1 13. 1 x dx 1 x  dx  28. 0 dx ex 1 e 2 x dx ex 1 1  29. 5 4 e  30. 1 3 31. 12 x  4 x 2  8dx  0 x5  x3 1 x 2 1  3 ln x ln x dx x 4 dx 32.  0 x 3  2 x 2  x dx 0 33. cos 2 x  2 3tgx cos 2 x dx cos 2 x  0  3 37.   0 x2 3 x3 x ln x  1 36. 38. (e x  1) 3 0  0 dx e x dx   2 2  cos 2 x 0  ln 2 ln 2 x ln 2 cos xdx 7 39. 34. 1  3 35. ln 3 2x  x(e  3 x  1)dx cos xdx 1  cos 2 x 2a dx 40.  x 2  a 2 dx 0 VI. MỘT SỐ TÍCH PHÂN ĐẶC BIỆT: Bµi to¸n më ®Çu: Hµm sè f(x) liªn tôc trªn [-a; a], khi ®ã: a a a 0  f ( x)dx   [ f ( x)  f ( x)]dx 3 3 ; VÝ dô: +) Cho f(x) liªn tôc trªn [- 2 2 ] tháa m·n f(x) + f(-x) = 2  2 cos2 x , 3 2 TÝnh:   f ( x)dx 3 2 1 +) TÝnh x 4  sin x dx 2  1 1  x a Bµi to¸n 1: Hµm sè y = f(x) liªn tôc vµ lÎ trªn [-a, a], khi ®ã:  2 1 VÝ dô: TÝnh:  ln(x  1 1  x )dx 2  cos x ln(x    f ( x)dx a = 0. 1  x 2 )dx 2 a Bµi to¸n 2: Hµm sè y = f(x) liªn tôc vµ ch½n trªn [-a, a], khi ®ã: a  f ( x)dx 0  f ( x)dx a =2  2 1 x x  cos x dx 4  sin 2 x  x dx  2 VÝ dô: TÝnh 1  x  1 Bµi to¸n 3: Cho hµm sè y = f(x) liªn tôc, ch½n trªn [-a, a], khi ®ã: a 4 2  a f ( x) a1  b x dx  0 f ( x)dx (1  b>0,  a)  2  3 x2 1 31  2 x dx VÝ dô: TÝnh:  sin x sin 3 x cos 5 x dx 1 ex 2  Bµi to¸n 4: NÕu y = f(x) liªn tôc trªn [0; 2 ], th×  2  sin VÝ dô: TÝnh 0  2  2 0 0  f (sin x)   f (cos x)dx  2 2009 sin x dx x  cos 2009 x  2009 0 sin x sin x  cos x dx  Bµi to¸n 5: Cho f(x) x¸c ®Þnh trªn [-1; 1], khi ®ã:   x 0 1  sin x dx VÝ dô: TÝnh b Bµi to¸n 6:  a 0 b  a  2  0 b f (b  x )dx   f ( x)dx 0  4 x sin x  1  cos x sin x  2  cos x dx b f (a  b  x)dx   f ( x )dx  0 xf (sin x)dx  2 0 f (sin x)dx x  sin 4 x ln(1  tgx )dx dx 0 VÝ dô: TÝnh Bµi to¸n 7: NÕu f(x) liªn tôc trªn R vµ tuÇn hoµn víi chu k× T th×: 0 a T  a T nT f ( x )dx   f ( x)dx 2008  VÝ dô: TÝnh  0  1  cos 2 x dx 0 C¸c bµi tËp ¸p dông:  4 1 1.  1 1 x dx 1 2x 2 2.   4 x7  x5  x3  x  1 dx cos 4 x  0 T f ( x )dx  n  f ( x)dx 0  2 1 dx 1 (1  e x )(1  x 2 ) 3. 1 2 5. 1 2  7. 1 x  cos 2 x ln(1  x )dx  4. sin 5 x 2   2 1  cos x x  cos x dx 2 x  4  sin  2 2 6. dx 8.  sin(sin x  nx)dx 0 tga cot ga e 1 e xdx 1 1  x 2   dx 1 x(1  x 2 ) (tga>0) VII. TÍCH PHÂN HÀM GIÁ TRỊ TUYỆT ĐỐI: 3 1. 2  x 2  1dx 2. 3 x 2  4 x  3 dx 0  2 2 3.  0  x x  m dx 0 4.  1  sin x dx 6.  3 4  sin 2 x dx 7.  4    ( x  2  x  2 )dx 2  3 11.  cos x  2 tg 2 x  cot g 2 x  2dx 6 2 8.  1  cos x dx 0 3 5 9. 2  3  5.  sin x dx 1 x 2  x dx 10. cos x  cos 3 x dx 12. 2 0 x  4 dx
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