Tich phan ham nhi phan thuc

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  • Gio vin: Nguyn Thnh Long Email: Loinguyen1310@gmail.com D: 01694 013 498

    1

    TCH PHN HM PHN THC, LNG GIC V M LOGARIT DI CON MT CA TCH PHN HM NH THC

    I. Trc khi tm hiu v chuyn ny chng ta tm hiu qua tch phn hm nh thc

    C dng ( )m n px a bx dx

    vi , , , , , , 0a b R m n p Q n p

    Ty thuc vo tnh cht v mi quan h qua li gia ly tha ca m, n, p m ta c cc cch t khc nhau.

    C th xt b ba s 1 1; ;m mp pn n

    TH 1: Nu p Z th ta t qx t vi q l mu s chung nh nht ca phn s ti gin ca m v n

    TH 2: Nu 1 , , , , , 1m sZ p r s Z r sn r

    ta t pnt a bx hoc nt a bx c bit

    - Nu rp Zs

    ta ch c t nt a bx

    - Nu rp Zs

    v 2,3,...p ta c th s dng tch phn tng phn, khi 2p TPTP mt ln, khi 3p

    TPTP hai ln,

    TH 3: Nu 1 , , ,m sp Z p r s Zn r

    th ta t n

    rn

    a bx tx

    Bi tp gii mu: TH 1: Nu p Z th ta t qx t vi q l mu s chung nh nht ca phn s ti gin ca m v n

    Bi 1: Tnh tch phn sau

    4

    1 1dxI

    x x

    Gii:

    Ta c

    114 41 2

    1 1

    11

    dxI x x dxx x

    Nhn xt: 11, , 1 22

    m n p Z q

    Cch 1:

    t 2

    2x t

    x tdx tdt

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    2

    i cn 4 21 1

    x tx t

    Khi

    2 2 2

    21 1 1

    21 1 42 2 2 2 ln ln 1 2 ln11 1 31

    t dtI dt t tt t t tt t

    Cch 2:

    t

    211

    2 1

    x tx t

    dx t dt

    i cn 4 31 2

    x tx t

    Khi

    2 3 3

    22 2 2

    1 31 1 42 2 2 2 ln 1 ln 2ln21 1 31

    t dt dtI dt t tt t t tt t

    TH 2: Nu 1 , , , , , 1m sZ p r s Z r sn r

    ta t pnt a bx hoc nt a bx c bit

    - Nu rp Zs

    ta ch c t nt a bx

    - Nu rp Zs

    v 2,3,...p ta c th s dng tch phn tng phn, khi 2p TPTP mt ln, khi 3p

    TPTP hai ln,

    Bi 2: (HDB A 2003 HNT 1996) Tnh tch phn sau 1

    3 2

    0

    1I x x dx

    Gii:

    Phn tch 1 1

    3 2 2 2

    0 0

    1 1 .I x x dx x x xdx

    Nhn xt: 1 13, 2, 22

    mm n pn

    Cch 1:

    t 2 2

    2 11x t

    t xxdx tdt

    i cn 1 00 1

    x tx t

    Khi 10 1 1

    2 2 2 2 2 4 3 5

    1 0 0 0

    1 1 21 13 5 15

    I t t dt t t dt t t dt t t

    Cch 2:

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    3

    t

    2

    21

    1

    2

    x tt x dtxdx

    i cn 1 00 1

    x tx t

    Khi 11 1 1 3 3 30 1 1

    2 2 2 2 2 2

    1 0 0 0

    1 1 1 1 2 2 21 12 2 2 2 3 3 15

    I t t dt t t dt t t dt t t

    Cch 4: t cos sinx t dx tdt

    Khi 2 2

    2 3 2 2

    0 0

    sin cos sin 1 sin cosI t tdt t t tdt

    Cch 4.1. t sin cost u tdt du Khi

    1 1 3 5

    2 2 2 4

    0 0

    1 2(1 )03 5 15

    u uI u u du u u du

    Cch 4.2.

    3 52 2

    2 2 2 4

    0 0

    sin sin 2sin 1 sin sin sin sin sin 23 5 150

    t tI t t d t t t d t

    .

    Cch 4.3.

    2 2 2 22

    0 0 0 0

    1 1 1 cos 4 1 1sin 2 cos cos cos cos 4 cos4 4 2 8 8

    tI t tdt tdt tdt t tdt

    Cch 5:

    1 12 2 2 2 2 2

    0 01 13 1

    2 2 2 22 2

    0 0

    1 11 1 1 1 1 12 2

    1 11 1 1 12 2

    I x x d x x x d x

    x d x x d x

    Cch 3: t 22dtt x xdx

    Bi 3: Tnh tch phn 7 3

    3 20 1

    x dxIx

    Gii :

    Cch 1: t

    2 3

    3 22

    11 3

    2

    x tt x

    xdx t dt

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    4

    i cn 2710

    txtx

    Khi

    3 27 2 22 5 24

    3 20 1 1

    1 . 2. 3 3 3 9312 2 2 5 2 101

    t t dtx xdx t tI t t dttx

    Cch 2:

    t

    2

    21

    1

    2

    x tt x dtxdx

    i cn 8710

    txtx

    Khi 2 1 5 28 8 3 3 3 3

    11 13

    1 81 1 1 3 312 2 2 5 2

    t dtI t t dt t t

    t

    Cch 3: Phn tch 23

    3 2 2 33 32 2

    1 11 1

    x xx x x x x xx x

    Cch 4: S dng tch phn tng phn

    t

    2

    2 32 2

    3 32 2

    21 31 1

    421 1

    u x du xdxd xx v xdv dx

    x x

    Bi 4: (HAN 1999) Tnh tch phn 4

    27 9

    dxIx x

    Gii: Phn tch

    4

    27 7

    4 11 2 29

    9x x dxdxI

    x x

    Nhn xt: 1 11, 2, 02

    mm n pn

    t 2 2

    2 99x t

    t xxdx tdt

    i cn 4 5

    47

    x ttx

    Khi 4 5 5

    2 22 24 47

    51 3 1 7ln ln46 3 6 4( 9) 99

    xdx tdt dt tItt t tx x

    Cch 2:

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    5

    t

    2

    29

    9

    2

    x tt x dtxdx

    Khi

    25

    116 2

    1 ...2

    9

    dtIt t

    n y liu ta c th lm c khng, c th bng cch t

    1 22

    2u t

    u tudu dt

    bn c gii tip nh

    Bi 5: (H KTQD 1997) Tnh tch phn sau: 1

    65 3

    0

    1I x x dx

    Gii:

    1 1

    6 65 3 3 3 2

    0 0

    1 1I x x dx x x x dx

    Nhn xt: 15, 3, 6 0mm n p Zn

    Cch 1:

    t 2

    3

    3

    1 31

    dt x dxt x

    x t

    i cn 1 00 1

    x tx t

    Khi 0 1 1 7 8

    6 6 6 7

    1 0 0

    1 1 1 1 11 13 3 3 3 7 8 168

    t tI t t dt t t dt t t dt

    Cch 2:

    1 1 1 16 6 6 75 3 2 3 3 2 3 2 3

    0 0 0 0

    7 83 31 16 73 3 3 3

    0 0

    1 1 1 1 1 1

    1 11 11 1 1 11 1 1 1 . .0 03 3 7 3 8 168

    I x x dx x x x dx x x dx x x dx

    x xx d x x d x

    Bi 6: (SGK T 112) Tnh tch phn sau 2

    2

    0

    1I x x dx

    Gii: Cch 1: S dng phng php tch phn tng phn

    t 2

    2

    2 11

    2

    du x dxu x

    xdv xdx v

    Khi 2 22 4 3

    2 2 3

    0 0

    2 2 341 1 6 60 02 4 3 3

    x x xI x x x dx x x dx

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    6

    Cch 2:

    t 1

    1x t

    t xdx dt

    i cn 2 30 1

    x tx t

    Khi

    3 3 4 3

    2 3 2

    1 1

    3 34114 3 3

    t tI t t dt t t dt

    Cch 3: S dng phng php phn tch Ta c 2 2 3 21 2 1 2x x x x x x x x

    Khi 2 4 3 2

    3 2

    0

    22 34204 3 2 3

    x x xI x x x dx

    Cch 4: S dng phng php a vo vi phn Ta c 2 2 3 21 1 1 1 1 1x x x x x x

    Khi 4 32 2 2 23 2 3 2

    0 0 0 0

    1 1 341 1 1 1 1 14 3 3

    x xI x dx x dx x d x x d x

    TH 3: Nu 1 , , ,m sp Z p r s Zn r

    th ta t n

    rn

    a bx tx

    Bi 7: Tnh tch phn sau 2

    4 21 1

    dxIx x

    Gii:

    Nhn xt: 1 12; 2; 22

    mm n p p Zn

    nn t 2

    22

    1x tx

    t

    222

    22

    22

    111

    1

    xtx t tdtx xdx

    t

    i cn 52

    21

    2

    x tx

    t

    Ta c

    5 322 2 2 32

    224 2 2

    61 1 2 52 2

    21 7 5 8 2. 1 53 2411 11 2

    tdx dx tdt tI t dt ttx x tx

    x

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    7

    Bi 8: Tnh tch phn sau:

    1

    31

    4

    31

    3

    dxxxxI .

    HD:

    Ta c

    1

    31

    3

    31

    2

    1.11 dxxx

    I 1 1

    3 2 3

    13

    1x x dx

    Nhn xt: 1 13, 2, 13

    mm n p Zn

    t 2 31 1

    2dt dxt

    x x . 6I bn c t gii

    Bi 9: Tnh tch phn sau 3

    2 33

    2(1 )

    dxIx

    Gii :

    Ta c 3 10; 2; 12

    mm n p p Zn

    t

    22 2

    22

    2 2

    11 1

    ( 1)

    xx tt tdtx xdx

    t

    i cn 2 33

    3332

    x t

    x t

    Khi 3 3 3

    22 2 2 2 23 2 3 2 342 222 3 3

    31 1

    2 31 2 3(1 ) 1 ( 1) . . .. . 3( 1)

    xdx tdt dtIt tx x t t tx tx x

    Bi tp t gii:

    Bi 1: (HSP II HN A 2000) Tnh tch phn 2

    31 1

    dxIx x

    HD:

    t 2

    323 3

    3112 1 1

    x dx dtt x dt dxtx x x

    Bi 2: (HAN A 1999) Tnh tch phn 4

    27

    1 7ln6 41

    dxIx x

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    8

    Bi 3: (HBKHN 1995) Tnh tch phn 2

    223

    121dxI

    x x

    Cch 1:

    t 2 22 2 2 21

    11 1 1x dx xdx dtt x dt dx

    tx x x x x

    v t tanu ,

    2 2u , 2 1

    dt dut

    .

    Cch 2: t 2

    1 , 0;cos 2 1

    dxt t dtt x x

    C1: t 1 cos

    xt

    vi

    20;t hoc

    tx

    sin1

    C2: t 2 1x t C3: t 2 1x t

    C4: t 1xt

    C5: Phn tch 2 21 1x x

    Bi 4: Tnh tch phn 1 3

    21

    01

    xI dxx

    C1: t tanx t C2: Phn tch 3 2 1x x x x

    C3: t

    2

    2 1

    u xxdv dx

    x

    C4: t x t C5: Phn tch 3 2 2 21 1 1x dx x xdx x d x

    Bi 5: (HTM 1997) Tnh tch phn 7 3

    3 20

    141201

    xI dxx

    Bi 6: (CKT KT I 2004) Tnh tch phn 2 4

    50 1

    xI dxx

    Bi 7: (C Hng hi 2007) Tnh tch phn 3

    3 2

    1

    14 315

    I x x dx

    Bi 8: (C S Phm Tin Giang 2006) Tnh tch phn 9

    3

    1

    468. 17

    I x x dx

    Bi 9: (C Nng Lm 2006) Tnh tch phn 1

    2

    0

    2 2 113

    I x x dx

    Bi 10: (C Ti Chnh K Ton IV 2005) Tnh tch phn 3

    3 5

    0

    8481.105

    I x x dx

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    9

    Bi 11: (C Khi A, B 2005) Tnh tch phn 1

    3 2

    0

    6 3 8. 35

    I x x dx

    Bi 12: (C GTVT 2005) Tnh tch phn 1

    5 2

    0

    81105

    I x x dx

    Bi 13: (H Hi Phng 2006) Tnh tch phn 1

    20

    1 ln 221

    xI dxx

    Bi 14: Tnh tch phn 1

    2 3

    0

    22 3 3 2 29

    I x x dx

    Bi 15: (C Dt may thi trang Tp.HCM 2007) Tnh tch phn

    3

    2 21

    313 121

    dxIx x

    Bi 16: Tnh tch phn 2 3

    2 23

    2 33 2 21

    dxIx x

    b. Tch phn hm phn thc, lng gic, m loga di con mt ca tch phn hm nh phn thc

    M rng pm nI u x a bu x d u x

    vi vi , , , , , , 0a b R m n p Q n p

    V c th ha trng hp 2 nh sau

    Nu 1 , , , , , 1m sZ p r s Z r sn r

    ta t pnt a bu x hoc

    nt a bu x

    c bit : Nu rp Zs

    ta ch c t nt a bu x

    Ta xt cc th d sau y

    Th d 1. (H DB B 2003) Tnh tch phn sau ln 5 2

    ln 2 1

    x

    x

    eI dxe

    Li gii.

    Ta c ln 5 ln 5 12

    2

    ln 2 ln 2

    11

    xx x x

    x

    eI dx e e dee

    th y chnh l tch phn nh thc vi

    1 11, 22

    mm n p Zn

    v xu x e

    t 2

    2 112

    xx

    x

    e te t

    e dx tdt

    i cn ln 5 2ln 2 1

    x tx t

    Khi

    22 22 3

    1 1

    1 2 22 202 2 1 21 13 3

    t tdtI t dt t t

    t

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    10

    Cch khc: t 1xe t

    Th d 2. (H B 2004 ) Tnh tch phn sau 1

    1 3ln .lne x xI dxx

    Li gii.

    Ta c 13

    1 1

    1 3ln .ln ln 1 3ln lne ex xI dx x x d x

    x

    th y chnh l tch phn nh thc vi

    1 11, 22

    mm n p Zn

    v lnu x x

    t

    2

    2

    1ln31 3ln

    23

    txx t

    dx tdtx

    i cn 2

    1 1x e tx t

    Khi 2 22 5 3

    2 4 2

    1 1

    22 1 2 2 116( )13 3 9 9 5 3 135

    t t tI t dt t t dt

    Cch khc: 1 3lnt x

    Th d 3. (PVBCTT 1999) Tnh tch phn sau 3 2

    1

    ln . 2 lne x xI dxx

    Li gii.

    Ta c 13 2

    2 3

    1 1

    ln . 2 ln ln 1 ln lne ex xI dx x x d x

    x

    th y chnh l tch phn nh thc vi

    1 11, 2, 13

    mm n p Zn

    v lnu x x

    t 3 2 23 ln2 ln2

    xt x t dt dxx

    i cn 3

    3

    31 2

    x e tx t

    Khi 3 3

    3 3

    3 3 33

    42 3

    32 2

    33 3 3 233 3 3. .

    2 2 2 42

    82tI t t dt t dt

    Cch khc: t 22 ln x t

    Th d 4. (H B 2010) Tnh tch phn sau 21

    ln2 ln

    e xI dxx x

    Li gii.

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    11

    Ta c

    2

    22

    1 1

    ln ln 2 ln ln2 ln

    e xI dx x x d xx x

    th y chnh l tch phn nh thc vi

    11, 1, 2 , 2mm n Z p Zn

    v lnu x x

    t ln 2

    2 lnx t

    t x dx dtx

    Khi 3 3

    2 22 2

    2 31 2 2ln2

    3 1ln2 3

    tI dt dt t

    t tt t

    Th d 5. (HDB 2002) Tnh tch phn sau

    ln 3

    30 1

    x

    x

    e dxIe

    Li gii.

    Ta c

    ln 3 ln 3 1

    33

    0 0

    11

    xx x

    x

    e dxI e dee

    th y chnh l tch phn nh thc vi

    1 10, 1, 12

    mm n p Zn

    v xu x e

    t 2 1 2 2x xt e tdt e dx dx tdt

    Khi 2

    32

    212 2. 2 12

    tdtItt

    Th d 6. Tnh tch phn sau 2

    5 31

    dxIx x

    Li gii.

    Ta c 2 2

    13 25 3

    1 1

    1dxI x x dxx x

    y l tch phn nh thc vi 3, 2, 1m n p Z

    t

    2

    21

    1

    2

    x tt x dt xdx

    i cn 2 51 2

    x tx t

    Ta c

    2 2

    3 2 4 21 1

    11 1

    xI dx dxx x x x

    Khi

    5 5

    2 22 2

    51 1 1 1 1 1 3 1 5ln ln 2 ln22 1 2 1 1 8 2 21 1

    dt tI dtt t t tt t t

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    12

    Th d 7. Tm nguyn hm:

    2

    391x dxI

    x

    Li gii.

    Ta c

    2

    39239 11

    x dxI x x dxx

    y l tch phn nh thc vi

    12, 1, 39 3mm n p Z Zn

    t 1 1t x x t dx dt Khi

    239 39 38 37 38 37 36

    1 1 1 1 1 1 2 1 1 1238 37 36

    t dtI dt dt dt C

    t t t t t t t

    vi 1t x

    Th d 8. (H B 2005) Tnh tch phn sau 2

    0

    sin 2 .cos1 cos

    x xI dxx

    Li gii. Phn tch

    22 2 2

    12

    0 0 0

    sin 2 .cos sin .cos2 2 cos 1 cos cos1 cos 1 cos

    x x x xI dx dx x x d xx x

    th y chnh l tch phn nh thc

    vi 2, 1, 1m n p Z v cosu x x

    t sin

    1 coscos 1dt xdx

    t xx t

    i cn 1

    220

    txtx

    Khi 21 2 2

    2 1

    1 212 2 2 2 2 ln 2 ln 2 112

    t tI dt t dt t tt t

    Th d 9. (HTS 1999) Tnh tch phn sau 2

    2

    0

    sin cos 1 cosI x x x dx

    Li gii.

    Ta c 2 2

    2 2

    0 0

    sin cos 1 cos cos 1 cos cosI x x x dx x x d x

    th y chnh l tch phn nh thc vi

    1, 1, 2m n p Z v cosu x x

    t sin

    1 coscos 1

    xdx dtt x

    x t

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    13

    i cn 1

    220

    txtx

    Khi 1 2 4 3

    2 3 2

    2 1

    2 17114 3 12

    t tI t t dt t t dt

    Nhn xt: Nu gp tch phn l tng (hiu) ca hai tch phn nh thc m c cng cch t th ta vn tnh nh trong l thuyt

    Th d 10. (H A 2005) Tnh tch phn sau 2

    0

    sin 2 sin1 3cos

    x xI dxx

    Li gii.

    Ta c

    1 2

    2 2 21 12 2

    0 0 0

    sin 2cos 12cos 1 3cos cos 1 3cos cos

    1 3cosI I

    x xI dx x x d x x d x

    x

    Nhn xt: y chnh l tng ca hai nh thc cosu x x vi 1I ta c 11 2mm n Z

    n

    v vi 2I

    ta c 10, 1 1mm n Zn

    .

    Vy chung qui li ta c th

    t

    2

    2

    1cos31 3cos

    sin 231 3cos

    txx t

    x dtdxx

    i cn 1

    220

    txtx

    Khi 2 2

    3

    1

    24 2 4 2 3419 9 27 9 27

    tI dt t t

    Th d 11. (HQG HCM B 1997) Tnh tch phn sau 2

    0

    sin 31 cos

    xI dxx

    Li gii.

    Ta c 32 2 2

    12

    0 0 0

    sin 3 3sin 4sin 4cos 1 1 cos cos1 cos 1 cos

    x x xI dx dx x x d xx x

    th y chnh l tng ca

    hai tch phn nh thc tch phn nh thc vi 12, 1, 1 3mm n p Z Zn

    v cosu x x nn ta

    t cos 1

    1 cossin

    x tt x

    dt xdx

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    14

    i cn 1

    220

    txtx

    Khi

    21 22

    2 1

    4 1 1 234 8 2 3ln 8 3ln 2 21

    tI dt t dt t t t

    t t

    kt thc bi vit ny mi cc bn t gii cc tch phn sau

    Bi 1: (HDB D 2005) Tnh tch phn sau 3 2

    1

    ln 7615ln 1

    e xI dxx x

    Bi 2: (HBK 2000) Tnh tch phn sau ln 2 2

    0

    2 231

    x

    x

    eI dxe

    Bi 3: (HHH 98) Tnh tch phn I = dxxx

    xe 1 ln1.

    ln 4 2 23

    Bi 4: (HDB 2 2006) Tnh tch phn sau 1

    3 2 ln 10 2 1131 2ln

    e xI dxx x

    Bi 5: (HCT 1999) Tnh tch phn sau 21

    ln 1 (ln 2 1)2ln 1

    e xI dxx x

    Bi 7: Tnh tch phn sau 32

    321

    log 427 ln 21 3ln

    e xI dxx x

    Bi 8: (HDB 2004) Tnh tch phn sau ln 8 ln 8

    2

    ln 3 ln 3

    1. 1. .x x x x xI e e dx e e e dx

    Bi 9: Tnh tch phn sau ln 5

    ln 2

    1

    1

    x x

    x

    e eI dx

    e

    Bi 10: (HVNH TPHCM D 2000) Tnh tch phn sau 2

    20

    sin 4 32 6 ln41 cos

    xI dxx

    Bi 11: Tnh tch phn sau 2 32

    0

    15sin 2 1 sin4

    I x x dx

    Bi 12: (H BCVT 1997) Tnh tch phn sau 32

    20

    sin cos1 cos

    x xI dxx

    Bi 13: Tnh tch phn 36

    0

    sin 3 sin 3 1 1 ln 21 cos3 6 3

    x xI dxx

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    15

    Bi 14: (HDB B 2004) Tnh tch phn sau 3

    30

    6ln2

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    Bi 15: Tm nguyn hm 3

    10 6 7 8 9

    1 1 3 1 3 1 1 16 7 8 9( 1) ( 1) ( 1) ( 1) ( 1)

    x dxI Cx x x x x

    Gp theo a ch Email: Loinguyen1310@gmail.com hoc a ch: Nguyn Thnh Long S nh 15 Khu ph 6 Phng ngc tro Th x bm sn Thnh ph thanh ha

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