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) = 4 x 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 x 2 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 x cos 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 xcos 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 xln 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 1 1 1 1 ]
( ) '( )sin sincosax axu fx du f x dxax axdv ax dx v cosax dxe e 11 ' '11 11 11 ] ] @ Dang 2: ( ) ln( ) fx ax dxt ln( )( )( )dxduu axxdv fx dxv fx dx ' '@ Dang 3: sin. 1 1 ]axaxe dxcosax Vidu1: ti nh 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+ + + + + Ti nh I11201 dxx+ b ng phng pha p i bin sTinh I2 = 1 22 20(1 )x dxx + bng phng pha p tng ph n : 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 x 29) 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 2sinxdx ex61. 203 sincos sin2xdx x ex62. +40) 1 ln(dx tgx63. +402) cos 2 (sinx x dx64. +202) cos 2 )( sin 1 (cos ) sin 1 (dxx xx x65. 22sin 2 sin 7x xdx66. 24 40cos (sin cos ) +x x x dx67. 2304sin1 cos +xdxx68. 223 cos . 5 cosxdx x 69. 222 sin . 7 sinxdx x 70.40cos2sinxdxx
71. 402sinxdx
V. TCH PHN HM V T:badx x f x R )) ( , (Trong R(x, f(x)) c cc dng: +) R(x, x ax a+) t x = a cos2t, t ]2; 0 [+) R(x, 2 2x a ) t x = t a sin hoc x = t a cos+) R(x, nd cxb ax++) t t = nd cxb ax+++) R(x, f(x)) = + + + x x b ax2) (1 Vi ( + + x x2) = k(ax+b)Khi t t = + + x x2, hoc t t = b ax +1+) R(x, 2 2x a +) t x = tgt a, t ]2;2[ +) R(x, 2 2a x ) t x = xacos, t}2{ \ ] ; 0 [ +) R( )1 2 in n nx x x ; ;...; Gi k = BCNH(n1; n2; ...; ni) t x = tk 1. +3 2524 x xdx2. 23221 x xdx3. + + +212125 12 4 ) 3 2 ( x x xdx4. +2131 x xdx5. +2122008dx x6. +2122008 x dx7. +102 21 dx x x8. 103 2) 1 ( dx x9. ++312 2211dxx x x10. +22011dxxx11. +103 2) 1 ( xdx12. 2203 2) 1 ( xdx13. +1021 dx x14. 220221 xdx x15. +202 cos 7cosxxdx16. 202cos cos sindx x x x17. +202cos 2cosxxdx18. ++20cos 3 1sin 2 sindxxx x19. +703 231 xdx x20. 302 310 dx x x21. +101 2xxdx22. + +10231 x xdx x23. + +721 1 2xdx24. dx x x+108 153 1 25. 205 6 3cos sin cos 1xdx x x26. +3 ln0 1xedx27. + + +1121 1 x x dx28. +2 ln021xxe dx e29. 14528 4 12 dx x x30.+edxxx x1ln ln 3 131. ++3023 51dxxx x32. dx x x x+ 402 3233. + +013 2) 1 ( dx x e xx34. +3 ln2 ln21 lnlndxx xx35. +3022cos3 2cos2 cos dxxtgxxx36. +2 ln03) 1 (xxe dx e37. +302 cos 2cosxxdx38. +202cos 1cosxxdx39. dxxx++7033240. +adx a x202 2VI. MT S TCH PHN C BIT:Bi ton m u: Hm s f(x) lin tc trn [-a; a], khi : + a aadx x f x f dx x f0)] ( ) ( [ ) (V d: +) Cho f(x) lin tc trn [-23;23 ] tha mn f(x) + f(-x) = x 2 cos 2 2 , Tnh: 2323) (dx x f+) Tnh ++11241sindxxx xBi ton 1: Hm s y = f(x) lin tc v l trn [-a, a], khi : aadx x f ) ( = 0.V d: Tnh:+ +112) 1 ln( dx x x+ +222) 1 ln( cosdx x x xBi ton 2: Hm s y = f(x) lin tc v chn trn [-a, a], khi : aadx x f ) ( = 2adx x f0) (V d: Tnh+ 112 41 x xdx x222cos4 sin+ x xdxxBi ton 3: Cho hm s y = f(x) lin tc, chn trn [-a, a], khi : +a aaxdx x f dxbx f0) (1) ( (1b>0, a)V d: Tnh: ++3322 11dxxx +2215 cos 3 sin sindxex x xxBi ton 4: Nu y = f(x) lin tc trn [0; 2], th 2020) (cos ) (sin dx x f x fV d: Tnh+202009 20092009cos sinsindxx xx+20cos sinsindxx xxBi ton 5: Cho f(x) xc nh trn [-1; 1], khi : 0 0) (sin2) (sin dx x f dx x xfV d: Tnh+0sin 1dxxx+0cos 2sindxxx xBi ton 6: +babadx x f dx x b a f ) ( ) ( b bdx x f dx x b f0 0) ( ) (V d: Tnh +02cos 1sindxxx x+40) 1 ln( 4 sindx tgx xBi ton 7: Nu f(x) lin tc trn R v tun hon vi chu k T th:
+ T T aadx x f dx x f0) ( ) ( T nTdx x f n dx x f0 0) ( ) (V d: Tnh 200802 cos 1 dx xCc bi tp p dng:1. +1122 11dxxx2. + + 4443 5 7cos1dxxx x x x3. + +112) 1 )( 1 ( x e dxx4. +222sin 4cosdxxx x5. +2121)11ln( 2 cos dxxxx6.dx nx) x sin(sin20+7. +225cos 1sindxxx8. 1) 1 ( 1cot1212+++ gaetgaex xdxxxdx (tga>0)VII. TCH PHN HM GI TR TUYT I:1. 3321dx x2. + 2023 4 dx x x3.10dx m x x4. 22sindx x5. dx x sin 16. +362 22 cotdx x g x tg7. 4342 sindx x8. + 20cos 1 dx x9. +52) 2 2 ( dx x x10. 304 2 dxx11. 323cos cos cosdx x x x12.2) 421x 3x 2dx +
13. 53( x 2 x 2)dx+ 14. 222121x 2dxx+ 15. 3x02 4dx 16. 01 cos2xdx+
17. 201 sinxdx+18. dx x x202 VIII. NG DNG CA TCH PHN:TNH DIN TCH HNH PHNGV d 1 : Tnh din tch hnh phng gii hn bi a/ th hm s y = x + x -1 , trc honh , ng thng x = -2 v ng thng x = 1 b/ th hm s y = ex +1 , trc honh , ng thng x = 0 v ng thng x = 1 c/ th hm s y = x3 - 4x , trc honh , ng thng x = -2 v ng thng x = 4 d/ th hm s y = sinx , trc honh , trc tung v ng thng x = 2V d 2 : Tnh din tch hnh phng gii hn bi a/ th hm s y = x + x -1 , trc honh , ng thng x = -2 v ng thng x = 1 b/ th hm s y = ex +1 , trc honh , ng thng x = 0 v ng thng x = 1 c/ th hm s y = x3 - 4x , trc honh , ng thng x = -2 v ng thng x = 4 d/ th hm s y = sinx , trc honh , trc tung v ng thng x = 2Bi 1 : Cho (p) : y = x2+ 1 v ng thng (d): y = mx + 2. Tm m din tch hnh phng gii hn bi hai ng trn c din tch nh nhtBi 2: Cho y = x4- 4x2 +m (c) Tm m hnh phng gii hn bi (c) v 0x c din tch pha trn 0x v pha di 0x bng nhauBi 3: Xc nh tham s m sao cho y = mx chia hnh phng gii hn bi ' 013yx ox xyC hai phn din tch bng nhauBi 4: (p): y2=2x chia hnh phng gii bi x2+y2 = 8 thnh hai phn.Tnh din tch mi phnBi 5: Cho a > 0Tnh din tch hnh phng gii hn bi '+++ +4242 2113 2aax ayaa ax xyTm a din tch ln nhtBi 6: Tnh din tch ca cc hnh phng sau:1) (H1):22xy 44xy4 2 '2) (H2) : 2y x 4x 3y x 3 +' +3) (H3):3x 1yx 1y 0x 0 '4)(H4):22y xx y ' 5) (H5):2y xy 2 x ' 6) (H6):2y x 5 0x y 3 0 + '+ 7)(H7):lnxy2 xy 0x ex 1'8) (H8) : 22y x 2xy x 4x ' +9)(H9):23 3y x x2 2y x + ' 10) (H10):2y 2y x 0x y 0 + '+ 11)' ) (2 : ) (: ) (Oxx y dx y C12)' 1 : ) (2 : ) (: ) (xy de y Cx13)' + 11 22x yx y 14)' + 0 3422y xx y 15) ' +00 2yy xx y16'+22112xyxy17' 3 , 0 ,22y y x yx y 18) ' e xexy x y,10 , ln19. ' 3;6cos1;sin12 2 x xxyxy20): y = 4x x2 ; (p) v tip tuyn ca (p) i qua M(5/6,6)21) ' + + 11 44 25 42x yx yx x y 22) ' + + 15 33 45 622x yx x yx x y 23) 'e xyxyx y01
24) '+ 5 / // 1 /2x yx y25) 'x yx y23 26)'+ 02 / / 32yx x y 27) ' + x yx y42228)'+ + + 15 42 222yx x yx x y29) '+ 7/ 1 /22x yx y
30) ' 1 ; 203x xyx y31) ' x xyx x y; 03cos 2 sin32) '+ + 023yxx y
33) '+ + 222x yx x y 34) ' + 4 ; 06 32 222x xx x yx x y35) '+ 6/ 6 5 /2yx x y36) ' 21 2222yx x yx y 37) '+ 2/ 2 3 /2yx x y 38)'+ + 1/ 6 5 /2x yx x y39)' + 22/ 2 3 /x yx x y40)'+ 3/ 3 4 /2yx x y 41) '1 xe ye yx42) ' 1 ; 06 22x xx x xy43) ' / // sin/x yx y44) ' 84 4222yx x yx y 45) ' + +00 1 2 222yy xx y 46) ' 0) (2 2 2 2 ax a x y47) '+ y xx y sin) 1 (248) ' 2/ 1 /2xx y49) ' 2/ 1 /2xy x32) '+ 0sin) 1 (2xx yy x33) ' 2 44422xyxy34) '0 ;121; 04yxxyxx35) ' x y xyyx3 ; 005236) ' +1662 22y xx y37) 'xyxyx y272722 38)' x yx y4) 4 (23 239)' 10 ,1010/ log /x xyx y40)'22x ayy ax (a>0)41)' + xx x yx y0sin242) ' 2 22) 1 ( 8 272x yx y43) x2/25+y2/9 = 1 v hai tip tuyn i qua A(0;15/4)44) Cho (p): y = x2 v im A(2;5) ng thng (d) i qua A c h s gc k .Xc nh k din tch hnh phng gii hn bi (p) v (d) nh nht45)' + 03 4 22 3yx x x yTNH TH TCH VT TH TRN XOAYCng thc:
[ ] dx x f Vba2) (
[ ] dy y f Vba2) ( Bi 1: Cho min D gii hn bi hai ng : x2 + x - 5 = 0 ; x + y - 3 = 0Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi 2: Cho min D gii hn bi cc ng :y x;y 2 x;y 0 Tnh th tch khi trn xoay c to nn do D quay quanh trc OyBi 3: Cho min D gii hn bi hai ng : 2y (x 2) v y = 4Tnh th tch khi trn xoay c to nn do D quay quanh:a) Trc Oxb) Trc OyBi 4: Cho min D gii hn bi hai ng : 2 24 ; 2 y x y x +.Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi 5: Cho min D gii hn bi cc ng : 221;1 2xy yx +Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi 6: Cho min D gii hn bi cc ng y = 2x2 v y = 2x + 4 Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi 7:Cho min D gii hn bi cc ng y = y2 = 4x v y = x Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi 8: Cho min D gii hn bi cc ng y =2 21.xe x ;y = 0 ;x= 1 ; x = 2ab 0 y) ( : ) ( x f y C ba x b x xyObaxy0 xO) ( : ) ( y f x C b y a y Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi 9: Cho min D gii hn bi cc ng y = xlnx ;y = 0 ; x = 1 ; x = e Tnh th tch khi trn xoay c to nn do D quay quanh trc OxBi10: Cho min D gii hn bi cc ng y = x ) 1 ln(3x +; y = 0 ; x = 1 Tnh th tch khi trn xoay c to nn do D quay quanh trc Ox1) ' 4) 2 (2yx yquay quanh trc a) 0x; b) 0y2) ' 44 ,2 2yx y x y quay quanh trc a) 0x; b) 0y3) ' +1 , 0 , 0112x x yxy quay quanh trc a) 0x; b) 0y4) ' 022yx x y quay quanh trc a) 0x; b) 0y5) ' e x xyx x y; 10ln .quay quanh trc a) 0x; 6) (D) '+ > 110 3) 0 (2yx yx x yquay quanh trc a) 0x;( H) nm ngoi y = x27)'x yx y2 quay quanh trc a) 0x; 8) Min trong hnh trn (x 4)2 + y2 = 1quay quanh trc a) 0x; b) 0y9)Min trong (E):14 92 2 + y xquay quanh trc a) 0x; b) 0y10) ' 1 0 ; , 10x xyxe yquay quanh trc0x;11) ' + x xyx x y;20sin cos4 4quay quanh trc0x;12) ' x yx y3 102 quay quanh trc0x;13) Hnh trn tm I(2;0) bn knh R = 1quay quanh trc a) 0x; b) 0y14) ' 2 ; 044x xxy quay quanh trc0x;15) ' 0 ; 021y xyx y quay quanh trc a) 0x; b) 0y
