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

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

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

    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