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Nguyễn Thanh Tuù Ñaïi soá 12   1. f(x) = x 2 – 3x + x 1 2. f(x) = 2 4 32 x x + 3. f(x) = 2 1 x x − 4. f(x) = 2 22 )1( x x − 5. f(x) = 4 3 xxx ++ 6. f(x) = 3 21 xx − 7. f(x) = x x 2 )1( − 8f(x) = 3 1 x x − 9. f(x) = 2 sin2 2 x 10. f(x) = tan 2 x 11. f(x) = cos 2 x 12. f(x) = (tanx – cotx) 2 13. f(x) = xx 22 cos.sin 1 15. f(x) = sin3x 16. f(x) = 2sin3xcos2x 17. f(x) = e x (e x – 1) 18. f(x) = e x (2 + ) cos 2 x e x− 19. f(x) = 2a x + 3 x 20. f(x) = e 3x+1  !"#$%&'(#$)#*'(+,-'$. #/0112334Tính I = ∫ dxxuxuf )(')].([ bằng cách đặt t = u(x)  Đặt t = u(x) dxxudt )('=⇒  I = ∫ ∫ = dttfdxxuxuf )()(')].([ 5.6# 7 1. ∫ − dxx )15( 2. ∫ − 5 )23( x dx 3. dxx ∫ − 25 4. ∫ −12x dx 5. ∫ + xdxx 72 )12( 6. ∫ + dxxx 243 )5( 7. xdxx .1 2 ∫ + 8. ∫ + dx x x 5 2 9. ∫ + dx x x 3 2 25 3 10. ∫ + 2 )1( xx dx 11. dx x x ∫ 3 ln 12. ∫ + dxex x 1 2 . 13. ∫ xdxx cossin 4 14. ∫ dx x x 5 cos sin 15. ∫ gxdxcot 16. ∫ x tgxdx 2 cos 17. ∫ x dx sin 18. ∫ x dx cos 19. ∫ tgxdx 20. ∫ dx x e x 21. ∫ − 3 x x e dxe 22. ∫ dx x e tgx 2 cos 23. ∫ − dxx .1 2 24. ∫ − 2 4 x dx 25. ∫ − dxxx .1 22 26. ∫ + 2 1 x dx 27. ∫ − 2 2 1 x dxx 28. ∫ ++ 1 2 xx dx 29. ∫ xdxx 23 sincos 30. dxxx .1 ∫ − 31. ∫ +1 x e dx 32. dxxx .1 23 ∫ + 8#/0119:1; Nếu u(x) , v(x) là hai hàm số có đạo hàm liên tục trên I ∫ ∫ −= dxxuxvxvxudxxvxu )(').()().()(').( Hay ∫ ∫ −= vduuvudv ( với du = u’(x)dx, dv = v’(x)dx) 7 Nguyễn Thanh Tuù Ñaïi soá 12 1. ∫ xdxx sin. 2. ∫ xdxxcos 3. ∫ + xdxx sin)5( 2 4. ∫ ++ xdxxx cos)32( 2 5. ∫ xdxx 2sin 6. ∫ xdxx 2cos 7. ∫ dxex x . 8. ∫ xdxln 9. ∫ xdxx ln 10. dxx ∫ 2 ln 11. ∫ x xdxln 12. ∫ dxe x 13. ∫ dx x x 2 cos 14. ∫ xdxxtg 2 15. ∫ dxxsin 16. ∫ + dxx )1ln( 2 17. ∫ xdxe x cos. 18. ∫ dxex x 2 3 19. ∫ + dxxx )1ln( 2 20. ∫ xdx x 2 21. ∫ xdxxlg 22. ∫ + dxxx )1ln(2 23. ∫ + dx x x 2 )1ln( 24. ∫ xdxx 2cos 2 <=$#$>' <'$<=$#$>'5?'(=)=$!@AB'(<'$=$CD.'(+,-'$.=&5E'7 1. 1 3 0 ( 1)x x dx+ + ∫ 2. 2 2 1 1 1 ( ) e x x dx x x + + + ∫ 2. 3 1 2x dx− ∫ 3. 2 1 1x dx+ ∫ 4. 2 3 (2sin 3 )x cosx x dx π π + + ∫ 5. 1 0 ( ) x e x dx+ ∫ 6. 1 3 0 ( )x x x dx+ ∫ 7. 2 1 ( 1)( 1)x x x dx+ − + ∫ F 2 3 1 (3sin 2 )x cosx dx x π π + + ∫ 9. 1 2 0 ( 1) x e x dx+ + ∫ 10. 2 2 3 1 ( )x x x x dx+ + ∫ 11. 2 1 ( 1)( 1)x x x dx− + + ∫ 12. 3 3 1 x 1 dx( ). − + ∫ 13. 2 2 2 -1 x.dx x + ∫ 14. 2 e 1 7x 2 x 5 dx x − − ∫ 15. x 2 5 2 dx x 2+ + − ∫ 16. 2 2 1 x 1 dx x x x ( ). ln + + ∫ 17. 2 3 3 6 x dx x cos . sin π π ∫ 18. 4 2 0 tgx dx x . cos π ∫ 19. 1 x x x x 0 e e e e dx − − − + ∫ 20. 1 x x x 0 e dx e e . − + ∫ 21. 2 2 1 dx 4x 8x+ ∫ 22. 3 x x 0 dx e e ln . − + ∫ 23. 2 0 dx 1 xsin π + ∫ 24. ∫ − ++ 1 1 2 )12( dxxx 25. ∫ −− 2 0 3 ) 3 2 2( dxxx 26. ∫ − − 2 2 )3( dxxx 27. ∫ − − 4 3 2 )4( dxx 28. dx xx ∫       + 2 1 32 11 29. ∫ − 2 1 3 2 2 dx x xx 30. ∫ e e x dx 1 1 31. ∫ 16 1 .dxx 32. dx x xx e ∫ −+ 2 1 752 33. dx x x ∫         − 8 1 3 2 3 1 4 #$%&'(#$)#GHI'#$B7 Nguyễn Thanh Tuù Ñaïi soá 12 1. 2 3 2 3 sin xcos xdx π π ∫ 2. 2 2 3 3 sin xcos xdx π π ∫ 3. 2 0 sin 1 3 x dx cosx π + ∫ 4. 4 0 tgxdx π ∫ 4. 4 6 cot gxdx π π ∫ 5. 6 0 1 4sin xcosxdx π + ∫ 6. 1 2 0 1x x dx+ ∫ 7. 1 2 0 1x x dx− ∫ 8. 1 3 2 0 1x x dx+ ∫ 9. 1 2 3 0 1 x dx x + ∫ 10. 1 3 2 0 1x x dx− ∫ 11. 2 3 1 1 1 dx x x + ∫ 12. 1 2 0 1 1 dx x+ ∫ 13. 1 2 1 1 2 2 dx x x − + + ∫ 14. 1 2 0 1 1 dx x + ∫ 15. 1 2 2 0 1 (1 3 ) dx x+ ∫ 16. 2 sin 4 x e cosxdx π π ∫ 17. 2 4 sin cosx e xdx π π ∫ 18. 2 3 2 3 sin xcos xdx π π ∫ 19. 2 1 2 0 x e xdx + ∫ 20. 2 sin 4 x e cosxdx π π ∫ 21. 2 4 sin cosx e xdx π π ∫ 22. 2 1 2 0 x e xdx + ∫ 23. 2 3 2 3 sin xcos xdx π π ∫ 24. 2 2 3 3 sin xcos xdx π π ∫ 25. 2 0 sin 1 3 x dx cosx π + ∫ 26. 4 0 tgxdx π ∫ 27. 4 6 cot gxdx π π ∫ 28. 6 0 1 4sin xcosxdx π + ∫ 29. 1 2 0 1x x dx+ ∫ 30. 1 2 0 1x x dx− ∫ 31. 1 3 2 0 1x x dx+ ∫ 32. 1 2 3 0 1 x dx x + ∫ 33. 1 3 2 0 1x x dx− ∫ 34. 2 3 1 1 1 dx x x + ∫ 35. 1 1 ln e x dx x + ∫ 36. 1 sin(ln ) e x dx x ∫ 37. 1 1 3ln ln e x x dx x + ∫ 38. 2ln 1 1 e x e dx x + ∫ 39. 2 2 1 ln ln e e x dx x x + ∫ 40. 2 2 1 (1 ln ) e e dx cos x+ ∫ 41. 2 1 1 1 x dx x+ − ∫ 42. 1 0 2 1 x dx x + ∫ 43. 1 0 1x x dx+ ∫ 44. 1 0 1 1 dx x x+ + ∫ 45. 1 0 1 1 dx x x+ − ∫ 46. 3 1 1x dx x + ∫ 46. 1 1 ln e x dx x + ∫ 51. 2 2 1 (1 ln ) e e dx cos x+ ∫ 52. 1 2 3 0 5x x dx+ ∫ 53. ( ) 2 4 0 sin 1 cosx xdx π + ∫ 126. ∫ + 32 5 2 4xx dx 54. 4 2 0 4 x dx− ∫ 56. 1 2 0 1 dx x+ ∫ 57. dxe x ∫ − + 0 1 32 58. ∫ − 1 0 dxe x 59. 1 3 0 x dx (2x 1)+ ∫ 60. 1 0 x dx 2x 1+ ∫ 61. 1 0 x 1 xdx− ∫ 62. 1 2 0 4x 11 dx x 5x 6 + + + ∫ 63. 1 2 0 2x 5 dx x 4x 4 − − + ∫ 64. 3 3 2 0 x dx x 2x 1+ + ∫ 65. Nguyễn Thanh Tuù Ñaïi soá 12 6 6 6 0 (sin x cos x)dx π + ∫ 66. 3 2 0 4sin x dx 1 cosx π + ∫ 67. 4 2 0 1 sin2x dx cos x π + ∫ 68. 2 4 0 cos 2xdx π ∫ 69. 2 6 1 sin2x cos2x dx sinx cosx π π + + + ∫ 70. 1 x 0 1 dx e 1+ ∫ . 71. dxxx )sin(cos 4 0 44 ∫ − π 72. ∫ + 4 0 2sin21 2cos π dx x x 73. ∫ + 2 0 13cos2 3sin π dx x x 74. ∫ − 2 0 sin25 cos π dx x x 75. 0 2 2 2 2 2 3 x dx x x − + + − ∫ 76. 1 2 1 2 5 dx x x − + + ∫ 77. 2 3 2 0 cos xsin xdx π ∫ 78. 2 5 0 cos xdx π ∫ 79. 4 2 0 sin4x dx 1 cos x π + ∫ 80. 1 3 2 0 x 1 x dx− ∫ 81. 2 2 3 0 sin2x(1 sin x) dx π + ∫ 82. 4 4 0 1 dx cos x π ∫ 83. e 1 1 lnx dx x + ∫ 84. 4 0 1 dx cosx π ∫ 85. e 2 1 1 ln x dx x + ∫ 86. 1 5 3 6 0 x (1 x ) dx− ∫ 87. 6 2 0 cosx dx 6 5sinx sin x π − + ∫ 88. 3 4 0 tg x dx cos2x ∫ 89. 4 0 cos sin 3 sin2 x x dx x π + + ∫ 90. ∫ + 2 0 22 sin4cos 2sin π dx xx x 91. ∫ −+ − 5ln 3ln 32 xx ee dx 92. ∫ + 2 0 2 )sin2( 2sin π dx x x 93. ∫ 3 4 2sin )ln( π π dx x tgx 94. ∫ − 4 0 8 )1( π dxxtg 95. ∫ + − 2 4 2sin1 cossin π π dx x xx 96. ∫ + + 2 0 cos31 sin2sin π dx x xx 97. ∫ + 2 0 cos1 cos2sin π dx x xx 98. ∫ + 2 0 sin cos)cos( π xdxxe x 99. ∫ −+ 2 1 11 dx x x 100. ∫ + e dx x xx 1 lnln31 101. ∫ + − 4 0 2 2sin1 sin21 π dx x x 102. 1 2 0 1 x dx− ∫ 103. 1 2 0 1 dx 1 x+ ∫ 104. 1 2 0 1 dx 4 x− ∫ 105. 1 2 0 1 dx x x 1− + ∫ 106. 1 4 2 0 x dx x x 1+ + ∫ 107. 2 0 1 1 cos sin dx x x π + + ∫ 108. 2 2 2 2 0 x dx 1 x− ∫ 109. 2 2 2 1 x 4 x dx− ∫ 110. 2 3 2 2 1 dx x x 1− ∫ 111. 3 2 2 1 9 3x dx x + ∫ 112. 1 5 0 1 (1 ) x dx x − + ∫ 113. 2 2 2 3 1 1 dx x x − ∫ 114. 2 0 cos 7 cos2 x dx x π + ∫ 115. 1 4 6 0 1 1 x dx x + + ∫ 116. 2 0 cos 1 cos x dx x π + ∫ 117. ∫ ++ − 0 1 2 22xx dx 118. ∫ ++ 1 0 311 x dx 119. ∫ − − 2 1 5 1 dx x xx 120. 8 2 3 1 1 dx x x + ∫ 121. 7 3 3 2 0 1 x dx x+ ∫ 122. 3 5 2 0 1x x dx+ ∫ 123. ln2 x 0 1 dx e 2+ ∫ 124. 7 3 3 0 1 3 1 x dx x + + ∫ 125. 2 2 3 0 1x x dx+ ∫ #$%&'(#$)#<=$#$>'J'(#$K'7 Nguyễn Thanh Tuù Ñaïi soá 12 Công thức tích phân từng phần : u( )v'(x) x ( ) ( ) ( ) '( ) b b b a a a x d u x v x v x u x dx= − ∫ ∫ Tch phân cc hm s d pht hin u v dv @ AL sin ( ) ax ax f x cosax dx e β α           ∫ Đặt ( ) '( ) sin sin cos ax ax u f x du f x dx ax ax dv ax dx v cosax dx e e = =           ⇒       = =                   ∫ @ AL87 ( )ln( )f x ax dx β α ∫ Đặt ln( ) ( ) ( ) dx du u ax x dv f x dx v f x dx  = =   ⇒   =   =  ∫ @ ALM7 sin .       ∫ ax ax e dx cosax β α Đặt ax ax sin sin cos u e du ae dx ax ax dv dx v dx ax cosax   = =   ⇒       = =             ∫ Ví dụ 1: tính các tích phân sau a/ 1 2 2 0 ( 1) x x e dx x + ∫ đặt 2 2 ( 1) x u x e dx dv x  =   =  +  b/ 3 8 4 3 2 ( 1) x dx x − ∫ đặt 5 3 4 3 ( 1) u x x dx dv x  =   =  −  c/ 1 1 1 1 2 2 2 1 2 2 2 2 2 2 2 2 0 0 0 0 1 (1 ) (1 ) 1 (1 ) dx x x dx x dx dx I I x x x x + − = = − = − + + + + ∫ ∫ ∫ ∫ Tính I 1 1 2 0 1 dx x = + ∫ bằng phương pháp đJi biến số Tính I 2 = 1 2 2 2 0 (1 ) x dx x+ ∫ bằng phương pháp từng phần : đặt 2 2 (1 ) u x x dv dx x =    =  +  53N1 1. 3 3 1 ln e x dx x ∫ 2. 1 ln e x xdx ∫ 3. 1 2 0 ln( 1)x x dx + ∫ 4. 2 1 ln e x xdx ∫ 5. 3 3 1 ln e x dx x ∫ 6. 1 ln e x xdx ∫ 7. 1 2 0 ln( 1)x x dx + ∫ 8. 2 1 ln e x xdx ∫ 9. 2 0 ( osx)sinxx c dx π + ∫ 10. 1 1 ( )ln e x xdx x + ∫ 11. 2 2 1 ln( )x x dx + ∫ 12. 3 2 4 tanx xdx π π ∫ 13. 2 5 1 ln x dx x ∫ 14. 2 0 cosx xdx π ∫ 15. 1 0 x xe dx ∫ 16. 2 0 cos x e xdx π ∫ Tính các tích phân sau 1) ∫ 1 0 3 . dxex x 2) ∫ − 2 0 cos)1( π xdxx 3) ∫ − 6 0 3sin)2( π xdxx 4) ∫ 2 0 2sin. π xdxx Nguyễn Thanh Tuù Ñaïi soá 12 5) ∫ e xdxx 1 ln 6) ∫ − e dxxx 1 2 .ln).1( 7) ∫ 3 1 .ln.4 dxxx 8) ∫ + 1 0 2 ).3ln(. dxxx 9) ∫ + 2 1 2 .).1( dxex x 10) ∫ π 0 .cos. dxxx 11) ∫ 2 0 2 .cos. π dxxx 12) ∫ + 2 0 2 .sin).2( π dxxxx 13) 2 5 1 lnx dx x ∫ 14) 2 2 0 xcos xdx π ∫ 15) 1 x 0 e sinxdx ∫ 16) 2 0 sin xdx π ∫ 17) e 2 1 xln xdx ∫ 18) 3 2 0 x sinx dx cos x π + ∫ 19) 2 0 xsinx cos xdx π ∫ 20) 4 2 0 x(2cos x 1)dx π − ∫ 21) 2 2 1 ln(1 x) dx x + ∫ 22) 1 2 2x 0 (x 1) e dx+ ∫ 23) e 2 1 (xlnx) dx ∫ 24) 2 0 cosx.ln(1 cosx)dx π + ∫ 25) 2 1 ln ( 1) e e x dx x + ∫ 26) 1 2 0 xtg xdx ∫ 27) ∫ − 1 0 2 )2( dxex x 28) ∫ + 1 0 2 )1ln( dxxx 29) ∫ e dx x x 1 ln 30) ∫ + 2 0 3 sin)cos( π xdxxx 31) ∫ ++ 2 0 )1ln()72( dxxx 32) ∫ − 3 2 2 )ln( dxxx <=$#$>'$.$O+P7 1. ∫ +− − 5 3 2 23 12 dx xx x 2. ∫ ++ b a dx bxax ))(( 1 3. ∫ + ++ 1 0 3 1 1 dx x xx 4. dx x xx ∫ + ++ 1 0 2 3 1 1 5. ∫ + 1 0 3 2 )13( dx x x 6. ∫ ++ 1 0 22 )3()2( 1 dx xx 7. ∫ + − 2 1 2008 2008 )1( 1 dx xx x 8. ∫ − +− ++− 0 1 2 23 23 9962 dx xx xxx 9. ∫ − 3 2 22 4 )1( dx x x 10. ∫ + − 1 0 2 32 )1( dx x x n n 11. ∫ ++ − 2 1 24 2 )23( 3 dx xxx x 12. ∫ + 2 1 4 )1( 1 dx xx 13. ∫ + 2 0 2 4 1 dx x 14. ∫ + 1 0 4 1 dx x x 15. dx xx ∫ +− 2 0 2 22 1 16. ∫ + 1 0 32 )1( dx x x 17. ∫ +− 4 2 23 2 1 dx xxx 18. ∫ +− ++ 3 2 3 2 23 333 dx xx xx 19. ∫ + − 2 1 4 2 1 1 dx x x 20. ∫ + 1 0 3 1 1 dx x 21. ∫ + +++ 1 0 6 456 1 2 dx x xxx 22. ∫ + − 1 0 2 4 1 2 dx x x 23. ∫ + + 1 0 6 4 1 1 dx x x 24. 1 2 0 4 11 5 6 x dx x x + + + ∫ Nguyễn Thanh Tuù Ñaïi soá 12 25. 1 2 0 1 dx x x+ + ∫ 26. ∫ − + 3 2 1 2 dx x x 27. dx x x ∫       − + − 1 0 3 1 22 28. ∫ −       +− − − 0 1 12 12 2 dxx x x 29. dxx x x ∫       −− + − 2 0 1 2 13 30. dx x xx ∫ + ++ 1 0 2 3 32 31. dxx x xx ∫ −         +− − ++ 0 1 2 12 1 1 32. dxx x xx ∫         +− + −+ 1 0 2 1 1 22 33. ∫ ++ 1 0 2 34xx dx D<=$#$>'$.Q%R'(()=7 1. xdxx 4 2 0 2 cossin ∫ π 2. ∫ 2 0 32 cossin π xdxx 3. dxxx ∫ 2 0 54 cossin π 4. ∫ + 2 0 33 )cos(sin π dxx 5. ∫ + 2 0 44 )cos(sin2cos π dxxxx 6. ∫ −− 2 0 22 )coscossinsin2( π dxxxxx 7. ∫ 2 3 sin 1 π π dx x 8. ∫ −+ 2 0 441010 )sincoscos(sin π dxxxxx 9. ∫ − 2 0 cos2 π x dx 10. ∫ + 2 0 sin2 1 π dx x 11. ∫ + 2 0 2 3 cos1 sin π dx x x 12. ∫ 3 6 4 cos.sin π π xx dx 13. ∫ −+ 4 0 22 coscossin2sin π xxxx dx 14. ∫ + 2 0 cos1 cos π dx x x 15. ∫ − 2 0 cos2 cos π dx x x 16. ∫ + 2 0 sin2 sin π dx x x 17. ∫ + 2 0 3 cos1 cos π dx x x 18. ∫ ++ 2 0 1cossin 1 π dx xx 19. ∫ − 2 3 2 )cos1( cos π π x xdx 20. ∫ − ++ +− 2 2 3cos2sin 1cossin π π dx xx xx 21. ∫ 4 0 3 π xdxtg 22. dxxg ∫ 4 6 3 cot π π 23. ∫ 3 4 4 π π xdxtg 24. ∫ + 4 0 1 1 π dx tgx 25. ∫ + 4 0 ) 4 cos(cos π π xx dx 26. ∫ ++ ++ 2 0 5cos5sin4 6cos7sin π dx xx xx 27. ∫ + π 2 0 sin1 dxx 28. ∫ ++ 4 0 13cos3sin2 π xx dx 29. ∫ + 4 0 4 3 cos1 sin4 π dx x x 30. ∫ + ++ 2 0 cossin 2sin2cos1 π dx xx xx 31. ∫ + 2 0 cos1 3sin π dx x x 32. ∫ − 2 4 sin2sin π π xx dx 33. ∫ 4 0 2 3 cos sin π dx x x 34. ∫ + 2 0 32 )sin1(2sin π dxxx 35. ∫ π 0 sincos dxxx 36. ∫ − 3 4 3 3 3 sin sinsin π π dx xtgx xx 37. ∫ ++ 2 0 cossin1 π xx dx 38. ∫ + 2 0 1sin2 π x dx 39. ∫ 2 4 53 sincos π π xdxx 40. ∫ + 4 0 2 cos1 4sin π x xdx 41. ∫ + 2 0 3sin5 π x dx 2. ∫ 6 6 4 cossin π π xx dx Nguyn Thanh Tuự ẹaùi soỏ 12 43. + 3 6 ) 6 sin(sin xx dx 44. + 3 4 ) 4 cos(sin xx dx 45. 3 4 6 2 cos sin x xdx 46. dxxtgxtg ) 6 ( 3 6 + 47. + 3 0 3 )cos(sin sin4 xx xdx 48. + 0 2 2 )sin2( 2sin x x 49. 2 0 3 sin dxx 50. 2 0 2 cos xdxx 51. + 2 0 12 .2sin dxex x 52. dxe x x x + + 2 0 cos1 sin1 53. + 4 6 2cot 4sin3sin dx xgtgx xx 54. + 2 0 2 6sin5sin 2sin xx xdx 55. 2 1 )cos(ln dxx 56. 3 6 2 cos )ln(sin dx x x 57. dxxx 2 0 2 cos)12( 58. 0 2 cossin xdxxx 59. 4 0 2 xdxxtg 60. 0 22 sin xdxe x 61. 2 0 3sin cossin 2 xdxxe x 62. + 4 0 )1ln( dxtgx 63. + 4 0 2 )cos2(sin xx dx 64. + 2 0 2 )cos2)(sin1( cos)sin1( dx xx xx 65. 2 2 sin 2 sin 7 x xdx 66. 2 4 4 0 cos (sin cos )+ x x x dx 67. 2 3 0 4sin 1 cos + x dx x 68. 2 2 3cos.5cos xdxx 69. 2 2 2sin.7sin xdxx 70. 4 0 cos 2 sin xdx x 71. 4 0 2 sin xdx D<=$#$>'$.DSP7 b a dxxfxR ))(,( Trong đó R(x, f(x)) có các dạng: +) R(x, xa xa + ) Đặt x = a cos2t, t ] 2 ;0[ +) R(x, 22 xa ) Đặt x = ta sin hoặc x = ta cos +) R(x, n dcx bax + + ) Đặt t = n dcx bax + + +) R(x, f(x)) = +++ xxbax 2 )( 1 Với ( ++ xx 2 ) = k(ax+b) Khi đó đặt t = ++ xx 2 , hoặc đặt t = bax + 1 +) R(x, 22 xa + ) Đặt x = tgta , t ] 2 ; 2 [ Nguyn Thanh Tuự ẹaùi soỏ 12 +) R(x, 22 ax ) Đặt x = x a cos , t } 2 {\];0[ +) R ( ) 1 2 i n n n x x x; ; ; Gọi k = BCNH(n 1 ; n 2 ; ; n i ) Đặt x = t k Bi tp vn dng 1. + 32 5 2 4xx dx 2. 2 3 2 2 1xx dx 3. +++ 2 1 2 1 2 5124)32( xxx dx 4. + 2 1 3 1xx dx 5. + 2 1 2 2008dxx 6. + 2 1 2 2008x dx 7. + 1 0 22 1 dxxx 8. 1 0 32 )1( dxx 9. + + 3 1 22 2 1 1 dx xx x 10. + 2 2 0 1 1 dx x x 11. + 1 0 32 )1( x dx 12. 2 2 0 32 )1( x dx 13. + 1 0 2 1 dxx 14. 2 2 0 2 2 1 x dxx 15. + 2 0 2cos7 cos x xdx 16. 2 0 2 coscossin dxxxx 17. + 2 0 2 cos2 cos x xdx 18. + + 2 0 cos31 sin2sin dx x xx 19. + 7 0 3 2 3 1 x dxx 20. 3 0 23 10 dxxx 21. + 1 0 12x xdx 22. ++ 1 0 2 3 1xx dxx 23. ++ 7 2 112x dx 24. dxxx + 1 0 815 31 25. 2 0 5 6 3 cossincos1 xdxxx 26. + 3ln 0 1 x e dx 27. +++ 1 1 2 11 xx dx 28. + 2ln 0 2 1 x x e dxe 29. 1 4 5 2 8412 dxxx 30. + e dx x xx 1 lnln31 31. dxxxx + 4 0 23 2 32. + + 3 0 2 35 1 dx x xx 33. ++ 0 1 3 2 )1( dxxex x 34. + 3ln 2ln 2 1ln ln dx xx x 35. + 3 0 2 2 cos 32 cos 2cos dx x tgx x x 36. + 2ln 0 3 )1( x x e dxe 37. + 3 0 2cos2 cos x xdx 38. + 2 0 2 cos1 cos x xdx 39. dx x x + + 7 0 3 3 2 40. + a dxax 2 0 22 D !"<=$#$>'GH=5T7 Bài toán mở đầu: Hàm số f(x) liên tục trên [-a; a], khi đó: += aa a dxxfxfdxxf 0 )]()([)( Ví dụ: +) Cho f(x) liên tục trên [- 2 3 ; 2 3 ] thỏa mãn f(x) + f(-x) = x2cos22 , Nguyn Thanh Tuự ẹaùi soỏ 12 Tính: 2 3 2 3 )( dxxf +) Tính + + 1 1 2 4 1 sin dx x xx Bài toán 1: Hàm số y = f(x) liên tục và lẻ trên [-a, a], khi đó: a a dxxf )( = 0. Ví dụ: Tính: ++ 1 1 2 )1ln( dxxx ++ 2 2 2 )1ln(cos dxxxx Bài toán 2: Hàm số y = f(x) liên tục và chẵn trên [-a, a], khi đó: a a dxxf )( = 2 a dxxf 0 )( Ví dụ: Tính + 1 1 24 1xx dxx 2 2 2 cos 4 sin + x x dx x Bài toán 3: Cho hàm số y = f(x) liên tục, chẵn trên [-a, a], khi đó: = + aa a x dxxfdx b xf 0 )( 1 )( (1 b>0, a) Ví dụ: Tính: + + 3 3 2 21 1 dx x x + 2 2 1 5cos3sinsin dx e xxx x Bài toán 4: Nếu y = f(x) liên tục trên [0; 2 ], thì = 2 0 2 0 )(cos)(sin dxxfxf Ví dụ: Tính + 2 0 20092009 2009 cossin sin dx xx x + 2 0 cossin sin dx xx x Bài toán 5: Cho f(x) xác định trên [-1; 1], khi đó: = 00 )(sin 2 )(sin dxxfdxxxf Ví dụ: Tính + 0 sin1 dx x x + 0 cos2 sin dx x xx Bài toán 6: =+ b a b a dxxfdxxbaf )()( = bb dxxfdxxbf 00 )()( Ví dụ: Tính + 0 2 cos1 sin dx x xx + 4 0 )1ln(4sin dxtgxx Bài toán 7: Nếu f(x) liên tục trên R và tuần hoàn với chu kì T thì: = + TTa a dxxfdxxf 0 )()( = TnT dxxfndxxf 00 )()( Ví dụ: Tính 2008 0 2cos1 dxx Các bài tập áp dụng: [...]... = dx 107/I = x 1 + 2x 1 (x + 1)(4 + 1) 1 1 2 108/I = 4 x cos xdx 0 1 0 2 4 x sin xdx 0 1 2 3 5 dx 124 /I = 2 dx x 2 4x 5 x 6x + 9 0 1 123 /I = 1 4 1 2x + 9 1 125 /I = dx 126 /I = dx 127 /I = 2 dx x +3 2x 2 + 8x + 26 x (x + 1) 5 0 1 Nguyn Thanh Tuự ẹaùi soỏ 12 0 sin 2x 1 1 x 3 4x dx 129 /I = 128 */I = (2 + sin x) 2 (x + 1)(x 2 + 3x + 2) dx 130/I = (x 3 + 1) dx 0 2 0 3 3 4sin 3 x 1 sin 3 x... 110*/I = 0 e ln x 1 111/I = e 2x sin 2 xdx 112/ I = x 2 ln(1 + )dx 113/I = (x + 1) 2 dx 1 x 0 1 2 e 1 2 3 2 t ln x 114/I = x.ln 1 + x dx 115/I = ữ dx I < 2 116/I = sin x.ln(cos x)dx 1 x 1 x 0 117/I = 0 e2 2 cos (ln x)dx 118/I = 1 4 1 119*/I = cos x dx 0 4 1 cos3 x dx 0 2 2 0 1 2 0 1 + cos 120 /I = x 3e x dx 121 /I = esin 2 x sin x cos3 xdx 122 /I = sin 2x dx 4 0 137/I = x 3 4 1 sin... 0 0 1 x + 2x + 1 286/I = 288/I = 1 2 0 1 1 1 (3 + 2x) 2 2 2 2 5 + 12x + 4x 2 2 dx 287/I = 0 1 x + 1+ x dx cos x + sin x 2 cos x 289/I = dx 290/I = (cos3 x + sin 3 x)dx dx 3 + sin 2x 2 + cos 2x 0 2 4 2 0 0 2 1 dx 2 + sin x 0 291/I = cos5 x sin 4 xdx 292/I = cos 2x(sin 4 x + cos 4 x)dx 293/I = Nguyn Thanh Tuự ẹaùi soỏ 12 2 1 sin 2 x 1 294/I = 308*/I = dx 309*/I = x dx 2 cos x dx 3... 310*/I = 313*/I = 1 2 2 2 tgx sin x sin 4 x 311/I = 312* /I = dx cos x + sin x dx cos 4 x + sin 4 x dx 2 0 1 ln (cos x) 0 0 2 1 1 1 sin x 314*/I = x 315*/I = e dx 2 dx 1 (e + 1)(x + 1) 0 cos x + sin x 0 x2 1 316*/I = x2 + 4 0 dx 318*/Tỡm x> 0 sao cho 317*/I = cos3 x cos 4 3cos 2 x + 3 dx 0 t e (t + 2) 2 dt = 1 0 320*/I = 3x 2 + 6x + 1dx 12 / I = 13*/ I = 3 2 x 4 1 3 x 2 4x 2 2 dx 35/I =... x 2 1 x 1cos 2 x ln(1 + x )dx 5 (1 + e 4 1 2 2 2 x + cos x dx 4 sin 2 x ẹaùi soỏ 12 4 dx = 1 (tana>0) x(1 + x 2 ) VII TCH PHN HM GI TR TUYT I: 3 1 x 2 2 1dx 2 3 5 1 sin x dx x 2 6 2 10 2 13 0 x 4 dx 11 16 2 ( x + 2 x 2 )dx 0 1 + cos 2xdx cos x 14 3 1 2 4dx 3 5 4 5 2 2 17 1 + cos x dx 0 0 12 x 2 3x + 2dx x 8 3 ( x + 2 x 2 )dx 2 2 sin 2 x dx 7 4 5 3 3 4 tg x + cot g... 194/I = 0 3 1 + 3cos x 4 1 2sin 2 0 x dx 195/I = 1 + sin 2x 1 + cos x 3 0 x 5 + 2x 3 2 x +1 1 x x2 196/I = dx dx 212/ I = dx 213/I = 2 4 x2 4 x2 cos x 1 + cos x 0 0 4 tgx 1 dx Nguyn Thanh Tuự 1 2 2 4 214/I = 15/I = sin 3x dx 216/I = x 2 1 dx cos x + 1 0 0 218/I = x ẹaùi soỏ 12 x3 7 3 0 2 1+ x 2 dx 219/I = ln 2 0 2 2 0 1 x2 dx 217/I = 1 + x 4 dx 1 2 x2 1 x2 1 1 1 ex dx 220/I = x 1 ... 9) Miền trong (E): x + y = 1 quay quanh trục a) 0x; b) 0y 9 4 y = xe ẽ 10) y = 0 quay quanh trục 0x; x = 1, ;0 x 1 y = cos 4 x + sin 4 x 11) y = 0 quay quanh trục 0x; x = ; x = 2 y = x2 12) quay quanh trục 0x; y = 10 3x 13) Hình tròn tâm I(2;0) bán kính R = 1 quay quanh trục a) 0x; b) 0y 4 14) y = quay quanh trục 0x; x4 x = 0; x = 2 y = x 1 15) y = 2 quay quanh trục a) 0x;... hn bi cỏc ng (C) : y = f ( x ) (C' ) : y = g( x ) x = a ; x = b b l S= f (x ) g(x ) dx a 1.Tớnh din tớch hỡnh phng gii hn bi: a) (C): y = 3x4 4x2 + 5 ; Ox ; x = 1; x = 2 Nguyn Thanh Tuự ẹaùi soỏ 12 2 b) (C): y = x x v (d): y = 4 4x ; Oy ; ng thng x = 3 c) y = sinx ; y = cosx ; x = 0; x = d) y = x2 x ; Ox d) y = (2 + cosx)sinx ; y = 0 ; x = /2 ; x = 3/2 e)y = x2 ; x + y + 2 = 0 f)x = y5 ;... xoay do hỡnh thang cong gii (C) : y = f ( x ) b 2 hn bi : Ox l V = [ f ( x )] dx a x = a; x = b 1.Tớnh th tớch hỡnh trũn xoay do cỏc hỡnh sau to thnh khi quay quanh trc Ox: Nguyn Thanh Tuự ẹaùi soỏ 12 a)y = sinx ; y = 0 ;x = 0 ; x = /2 b) y = cos x ; y = 0 ;x = 0 ; x = /4 c)y = ; y = 0 ; x = 0 ; x = /2 d)y = ; y = 0 ; x = /4; x = /2 e)y = xex ; y = 0 ;x = 0 ; x = 1 f)y= lnx ; y = 0 ; x =1 ; x = e... l ng thng i qua im M(1;1) cú h s gúc k < 0 ,(d) ln lt ct Ox v Oy ti A v B a)Tớnh th tớch vt th trũn xoay do tam giỏc OAB to thnh khi quay quanh Ox b)Tỡm k th tớch y nh nht 2 Nguyn Thanh Tuự ẹaùi soỏ 12 4 3 4/I = 3tg 2 x dx 5/I = (2cotg 2 x + 5) dx 6/I = 4 8/I = 6 1 + cos x dx 7/ I = 0 2 sin2 x.cos2xdx 0 x) 4 2 x-3sin2 x)dx 9 / I = (2cos dx sin( + x) 2 4 2 3 0 3 10 / I = 20/ I . ∫ − − 2 1 5 1 dx x xx 120 . 8 2 3 1 1 dx x x + ∫ 121 . 7 3 3 2 0 1 x dx x+ ∫ 122 . 3 5 2 0 1x x dx+ ∫ 123 . ln2 x 0 1 dx e 2+ ∫ 124 . 7 3 3 0 1 3 1 x dx x + + ∫ 125 . 2 2 3 0 1x x dx+ ∫ . dxx ∫ − 25 4. ∫ −12x dx 5. ∫ + xdxx 72 )12( 6. ∫ + dxxx 243 )5( 7. xdxx .1 2 ∫ + 8. ∫ + dx x x 5 2 9. ∫ + dx x x 3 2 25 3 10. ∫ + 2 )1( xx dx 11. dx x x ∫ 3 ln 12. ∫ + dxex x 1 2 . 13. ∫ xdxx. + ∫ Nguyễn Thanh Tuù Ñaïi soá 12 25. 1 2 0 1 dx x x+ + ∫ 26. ∫ − + 3 2 1 2 dx x x 27. dx x x ∫       − + − 1 0 3 1 22 28. ∫ −       +− − − 0 1 12 12 2 dxx x x 29. dxx x x ∫       −− + − 2 0 1 2 13

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