# Bai Tap Nguyen Ham Tich Phan

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I. Tm nguyn hm bng nhngha v cc tnh cht1/ Tm nguyn hm ca cc hm s.1.f(x) = x2 3x + x1S.F(x) =C xx x+ + ln2332 3

2.f(x) =243 2xx + S.F(x) =Cxx+ 3323 .f(x) = 21xx S. F(x) = lnx + x1 + C 4.f(x) = 22 2) 1 (xx S.F(x) = Cxxx+ + 12335. f(x) = 4 3x x x + + S.F(x) = Cx x x+ + +5443324534236.f(x) = 32 1x x S.F(x) =C x x + 3 23 27.f(x) = xx2) 1 ( S. F(x) = C x x x + + ln 48.f(x) = 31xx S.F(x) = C x x + 32359. f(x) = 2sin 22 x S.F(x) = x sinx + C10.f(x) = tan2x S.F(x) = tanx x + C11.f(x) = cos2x S.F(x) = C x x + + 2 sin4121

12.f(x) = (tanx cotx)2 S. F(x) = tanx - cotx 4x + C13. f(x) = x x2 2cos . sin1 S.F(x) =tanx- cotx + C14.f(x) = x xx2 2cos . sin2 cosS.F(x) =- cotx tanx + C 15.f(x) = sin3x S. F(x) = C x + 3 cos31

16.f(x) =2sin3xcos2x S.F(x) = C x x + cos 5 cos5117. f(x) = ex(ex 1) S.F(x) = C e ex x+ 221 18.f(x) = ex(2 +)cos2xex S.F(x) = 2ex + tanx + C19.f(x) = 2ax + 3xS.F(x) =Caax x+ +3 ln3ln2

20.f(x) = e3x+1S.F(x) = C e x++1 3312/ Tm hm s f(x) bit rng 1. f(x) = 2x + 1 v f(1) = 5 S. f(x) = x2 + x + 32.f(x) = 2 x2 v f(2) = 7/3 S. f(x) = 1323+ xx 3.f(x) = 4x x v f(4) = 0S. f(x) = 3402 382 x x x4. f(x) = x - 212 +x v f(1) = 2S.f(x) = 232122 + + xxx 5. f(x) = 4x3 3x2 + 2v f(-1) = 3 S.f(x) = x4 x3 + 2x + 36. f(x) = ax + 2 ) 1 ( , 4 ) 1 ( , 0 ) 1 ( ' ,2 f f fxb S. f(x) = 25 122+ +xxII. MT S PHNG PHP TM NGUYN HM1.Phng php i bin s.TnhI = dx x u x u f ) ( ' )]. ( [ bng cch t t = u(x)t t = u(x)dx x u dt ) ( ' I = dt t f dx x u x u f ) ( ) ( ' )]. ( [BI TPTm nguyn hm ca cc hm s sau:1. dx x ) 1 5 (2.5) 2 3 ( xdx3.dx x2 54.1 2xdx5.+ xdx x7 2) 1 2 ( 6.+ dx x x2 4 3) 5 (7.xdx x . 12+ 8.+dxx x529.+dxxx322 53 10. +2) 1 ( x xdx11. dxx x3ln12.+dx e xx 12.13.xdx xcos sin4 14.dxxx5cossin15.gxdx cot16.xtgxdx2cos17.xdxsin18.xdxcos 19.tgxdx20.dxxex21.3xxe dx e 22.dxxetgx2cos 23. dx x . 12 24. 24 xdx25. dx x x . 12 226.+21 xdx 27.221 xdx x28.+ + 12x xdx29.xdx x2 3sin cos 30.dx x x . 1 31.+1xedx 32.dx x x . 12 3+2. Phng php ly nguyn hm tng phn.Nu u(x) , v(x) l hai hm s c o hm lin tc trn I dx x u x v x v x u dx x v x u ) ( ' ). ( ) ( ). ( ) ( ' ). (Hay vdu uv udv ( vi du = u(x)dx,dv = v(x)dx)Tm nguyn hm ca cc hm s sau:1. xdx x sin .2.xdx x cos 3.+ xdx x sin ) 5 (2 4+ + xdx x x cos ) 3 2 (25.xdx x 2 sin6.xdx x 2 cos 7.dx e xx. 8. xdx ln9.xdx x ln 10.dx x2ln11.xxdx ln12.dx ex13. dxxx2cos14.xdx xtg215.dx x sin16.+ dx x ) 1 ln(217.xdx excos . 18.dx e xx23 19.+ dx x x ) 1 ln(2 20.xdxx221.xdx xlg 22.+dx x x ) 1 ln( 2 23.+dxxx2) 1 ln( 24.xdx x 2 cos2TCH PHNI. TNH TCH PHN BNG CCH S DNG TNH CHT V NGUYN HM C BN:1.130( 1) x x dx + +2.2211 1( )ex x dxx x+ + +

2. 312 x dx 3. 211 x dx +

4. 23(2sin 3 ) x cosx x dx+ +5. 10( )xe x dx +

6. 130( ) x x x dx + 7.21( 1)( 1) x x x dx + + 8. 231(3sin 2 ) x cosx dxx+ +9. 120( 1)xe x dx + +

10. 22 31( ) x x x x dx + +11.21( 1)( 1) x x x dx + + 12.331x 1 dx ( ).+13. 222-1x.dxx+14. 2e17x 2 x 5dxx 15. x 252dxx 2 + + 16. 221x 1 dxx x x( ).ln++17. 2 336x dxxcos .sin18. 420tgx dxx.cos19. 1x xx x0e ee edx+20. 1xx x0e dxe e.+21. 221dx4x 8x +22. 3x x0dxe eln.+22. 20dx1 x sin+24. + +112) 1 2 ( dx x x 25. 203)322 ( dx x x

26. 22) 3 ( dx x x27. 432) 4 ( dx x28.dxx x

,`

.|+213 21 129. 21322dxxx x 30. eexdx1131. 161.dx x32.dxxx xe +217 5 2 33. dxxx

,`

.|813 2314II. PHNG PHP T N PH:1. 23 23sin xcos xdx 2. 22 33sin xcos xdx3. 20sin1 3 xdxcosx+3. 40tgxdx

4. 46cot gxdx5. 601 4sin xcosxdx+6. 1201 x x dx + 7. 1201 x x dx 8. 13 201 x x dx +9. 1 2301xdxx + 10. 13 201 x x dx 11. 23111 dxx x +12. 12011dxx + 13. 12112 2dxx x+ +

14. 12011dxx +15. 12 201(1 3 ) dxx +16. 2sin4xe cosxdx 17. 24sincosxe xdx

18. 2120xe xdx+ 19. 23 23sin xcos xdx

20. 2sin4xe cosxdx 21. 24sincosxe xdx 22. 2120xe xdx+23. 23 23sin xcos xdx 24. 22 33sin xcos xdx25. 20sin1 3 xdxcosx+

26. 40tgxdx27. 46cot gxdx

28. 601 4sin xcosxdx+29. 1201 x x dx +30. 1201 x x dx 31. 13 201 x x dx +32. 1 2301xdxx + 33. 13 201 x x dx 34. 23111 dxx x +35. 11 lnexdxx+36. 1sin(ln )exdxx37. 11 3ln lnex xdxx+38. 2ln 11e xedxx+39. 221 lnlneexdxx x+

40. 221(1 ln )eedxcos x +41. 211 1xdxx + 42. 102 1xdxx +43. 101 x x dx +44. 1011dxx x + + 45. 1011dxx x + 46. 311 xdxx+ 46.11 lnexdxx+ 47. 1sin(ln )exdxx48. 11 3ln lnex xdxx+ 49. 2ln 11e xedxx+50. 221 lnlneexdxx x+

51. 221(1 ln )eedxcos x +52. 12 305 +x x dx53. ( )240sin 1 cos +x xdx54. 4204 x dx 55. 4204 x dx 56. 1201dxx + 57. dx ex+013 258. 10dx ex59. 130xdx(2x 1) + 60. 10xdx2x 1 +

61. 10x 1 xdx 62. 1204x 11dxx 5x 6++ +

63. 1202x 5dxx 4x 4 +64.3 320xdxx 2x 1 + + 65.66 60(sin x cos x)dx+ 66.3 204sin xdx1 cosx+ 67.4201 sin2xdxcos x+ 68.240cos 2xdx

69. 261 sin2x cos2xdxsinx cosx+ ++ 70.1x01dxe 1 +. 71.dx x x ) sin (cos404 4 72. +40 2 sin 2 12 cosdxxx 73. +20 1 3 cos 23 sindxxx 74. 20 sin 2 5cosdxxx

75. ++0223 22 2dxx xx76.+ + 1125 2x xdx

77. 23 20cos xsin xdx 78.250cos xdx79. 420sin4xdx1 cos x+80. 13 20x 1 x dx 81. 22 30sin2x(1 sin x) dx+82. 4401dxcos x

83. e11 lnxdxx+84. 401dxcosx85. e 211 ln xdxx+ 86.15 3 60x (1 x ) dx 87. 620cosxdx6 5sinx sin x +88. 3 40tg xdxcos2x89. 40cos sin3 sin2x xdxx++ 90.+202 2sin 4 cos2 sindxx xx 91. +5 ln3 ln 3 2x xe edx 92. +202) sin 2 (2 sindxxx 93. 342 sin) ln(dxxtgx 94.408) 1 (dx x tg

95.+242 sin 1cos sindxxx x96.++20 cos 3 1sin 2 sindxxx x 97. +20 cos 1cos 2 sindxxx x98.+20sincos ) cos (xdx x ex

99. +21 1 1dxxx 100. +edxxx x1ln ln 3 1

101. +4022 sin 1sin 2 1dxxx102. 1201 x dx 103. 1201dx1 x + 104. 1201dx4 x

105. 1201dxx x 1 +106.14 20xdxx x 1 + +

107.2011 cos sindxx x+ + 108. 22 220xdx1 x 109. 22 21x 4 x dx 110. 23221dxx x 1

101. 3 2219 3xdxx+ 112. 1501(1 )xdxx+

113. 222311dxx x114. 20cos7 cos2xdxx+

115. 1 46011 xdxx++ 116. 20cos1 cosxdxx+

117.+ + 0122 2x xdx118. + +10 3 1 1 xdx

119.21 51dxxx x120.82311dxx x+ 121. 7 33 201xdxx + 122. 35 201 x x dx +

123. ln2x01dxe 2 +124. 733013 1xdxx++

125. 22 301 x x dx +126.+3 2524 x xdx II. PHNG PHP TCH PHN TNG PHN: Cng thc tch phn tng phn : u( )v'(x) x ( ) ( ) ( ) '( )b bbaa ax d u x v x v x u x dx Tich phn cac ha m s d phat hin u va dv @ Dang 1 sin( )axaxfx cosax dxe ] ] ] ] ]

( ) '( )sin sincosax axu fx du f x dxax axdv ax dx v cosax dxe e ]] ' ']] ]] ]] ]] @ Dang 2: ( ) ln( ) fx ax dxt ln( )( )( )dxduu axxdv fx dxv fx dx ' '@ Dang 3: sin. ] ] ]axaxe dxcosax Vi du1: tinh cac tich phn sau a/1 220( 1)xx edxx + t 22( 1)xu x edxdvx '+b/3 84 32( 1)x dxx t 534 3( 1)u xx dxdvx 'c/1 1 1 1 2 2 21 2 2 2 2 2 2 2 20 0 0 01(1 ) (1 ) 1 (1 )dx x x dx x dxdx I Ix x x x+ + + + + Tinh I11201 dxx+ bng phng phap i bin sTinh I2 = 1 22 20(1 )x dxx + bng phng phap tng phn : t 2 2(1 )u xxdv dxx'+Bi tp1. 331lnexdxx2. 1lnex xdx3. 120ln( 1) x x dx +4. 21lnex xdx 5. 331lnexdxx 6. 1lnex xdx 7. 120ln( 1) x x dx + 8. 21lnex xdx9. 20( osx) sinx x c dx+10. 11( ) lnex xdxx+11. 221ln( ) x x dx +12. 324tan x xdx13.251ln xdxx14.20cos x xdx 15. 10xxe dx 16. 20cosxe xdxTnh cc tch phn sau 1) 103. dx e xx 2)20cos ) 1 (xdx x 3)603 sin ) 2 (xdx x4) 202 sin .xdx x

5)exdx x1ln6)edx x x12. ln ). 1 ( 7)31. ln . 4 dx x x8) +102). 3 ln( . dx x x9)+212. ). 1 ( dx e xx10)0. cos . dx x x 11)202. cos .dx x x 12)+202. sin ). 2 (dx x x x 13) 251lnxdxx14) 220xcos xdx 15) 1x0e sinxdx 16) 20sin xdx 17) e21xln xdx18) 320x sinxdxcos x+19) 20xsinxcos xdx20) 420x(2cos x 1)dx 21) 221ln(1 x)dxx+22) 12 2x0(x 1) e dx + 23) e21(xlnx) dx24) 20cosx.ln(1 cosx)dx+ 25) 21ln( 1)eexdxx+26) 120xtg xdx27)102) 2 ( dx e xx28) +102) 1 ln( dx x x29) edxxx1ln30)+203sin ) cos (xdx x x 31) + +20) 1 ln( ) 7 2 ( dx x x 32)322) ln( dx x x III. TCH PHN HM HU T:1. + 5322 31 2dxx xx2. + +badxb x a x ) )( (13. ++ +10311dxx x x4. dxxx x++ +1023115. +1032) 1 3 (dxxx6. + +102 2) 3 ( ) 2 (1dxx x7. +2120082008) 1 (1dxx xx8. + + + 0122 32 39 9 6 2dxx xx x x9. 322 24) 1 (dxx x10. + 1023 2) 1 (dxxxnn11. + +212 42) 2 3 (3dxx x xx12. +214) 1 (1dxx x13. +20241dxx14. +1041dxxx15. dxx x+ 2022 2116. +103 2) 1 (dxxx17. + 422 321dxx x x18. + + +32322 33 3 3dxx xx x19. +214211dxxx20. +10311dxx21. ++ + +1064 5 612dxxx x x22. +102412dxxx23. ++106411dxxx24. 1204 115 6xdxx x++ +25. 1201dxx x + +26. +3212dxxx 27. dxxx

,`

.|+10312 228.

,`

.|+ 011 21 22dx xxx

29. dx xxx

,`

.| +20121 3 30. dxxx x++ +10233 2 31. dx xx x x

,`

.|+ + +0121 21132. dx xxx x

,`

.|+ + +102112 2 33. + +1023 4x xdx IV. TCH PHN HM LNG GIC:1. xdx x4202cos sin2. 203 2cos sinxdx x3. dx x x205 4cos sin4. +203 3) cos (sindx x5. +204 4) cos (sin 2 cosdx x x x6. 202 2) cos cos sin sin 2 (dx x x x x7. 23sin1dxx8. +204 4 10 10) sin cos cos (sindx x x x x9. 20cos 2xdx10. +20sin 21dxx11. +2023cos 1sindxxx12. 364cos . sinx xdx13. +402 2cos cos sin 2 sinx x x xdx14. +20cos 1cosdxxx15. 20cos 2cosdxxx16. +20sin 2sindxxx17. +203cos 1cosdxxx18. + +201 cos sin1dxx x19. 232) cos 1 (cosxxdx20. + ++ 223 cos 2 sin1 cos sindxx xx x21. 403xdx tg22. dx x g463cot23. 344xdx tg24. +4011dxtgx25. +40)4cos( cosx xdx26. + ++ +205 cos 5 sin 46 cos 7 sindxx xx x27. + 20sin 1 dx x28. + +4013 cos 3 sin 2x xdx29. +4043cos 1sin 4dxxx30. ++ +20cos sin2 sin 2 cos 1dxx xx x31. +20cos 13 sindxxx32. 24sin 2 sinx xdx33. 4023cossindxxx34. +203 2) sin 1 ( 2 sindx x x35. 0sin cos dx x x36. 3433 3sinsin sindxxtgx x x37. + +20cos sin 1x xdx38. +201 sin 2xdx39. 245 3sin cosxdx x40. +402cos 14 sinxxdx41. +203 sin 5xdx2. 664cos sinx xdx43. +36)6sin( sinx xdx4. +34)4cos( sinx xdx45. 3462cossinxxdx46. dx x tgxtg )6(36+47. +303) cos (sinsin 4x xxdx48. +022) sin 2 (2 sinxx49. 203sindx x50. 202cosxdx x51. +201 2. 2 sindx e xx52. dx exxx++20cos 1sin 153. +462 cot4 sin 3 sindxx g tgxx x54. + 2026 sin 5 sin2 sinx xxdx55. 21) cos(ln dx x 56. 362cos) ln(sindxxx57. dx x x202cos ) 1 2 (58. 02cos sin xdx x x59. 402xdx xtg60. 02 2sin xdx ex61. 203 sincos sin2xdx x ex62. +40) 1 ln(dx tgx63. +402) cos 2 (sinx

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