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Phan Loai Mot So Dang Tich Phan Dac Biet

Feb 10, 2018

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  • 7/22/2019 Phan Loai Mot So Dang Tich Phan Dac Biet

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    http://ebooktoan.com 1TCH PHN

    I.CC PHNG PHP TNH TCH PHN

    1. Phng php i bin s

    Bi ton:Tnh ( )

    b

    a

    I f x dx ,*Phng php i bin dng I

    nh l. Nu 1) Hm ( )x u t c o hm lin tc trn on ; ,

    2) Hm hp ( ( ))f u t c xc nh trn ; ,

    3) ( ) , ( )u a u b ,

    th '( ) ( ( )) ( )b

    a

    I f x dx f u t u t dt

    .V d 1. Hy tnh cc tch phn sau:

    a)

    1

    2 3

    0

    5I x x dx b) 2

    4

    0

    sin 1 cosJ x xdx

    Gii: a) Ta c 3 25 3 t x dt x dx

    Khi x=0 th t=5

    Khi x=1 th t=6

    1 6

    2 3

    0 5

    53

    dt

    I x x dx t

    16 121

    2

    5

    6 61 1 ( ) 21 5 53 3 91

    2

    t

    t dt t t

    4 106 5

    3 9 .

    b) Ta c

    2

    4

    0

    (sin 1) (sin )J x d x

    51 6

    sin sin 25 5

    0x x

    V d 2. Hy tnh cc tch sau:

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    a)

    4

    2

    0

    4 x dx b)1

    2

    01

    dx

    x

    Gii: a) t 2sin , ;2 2

    x t t

    .

    Khi x = 0 th t = 0. Khi 2x th2

    t .

    T 2sinx t 2cosdx tdt 4 2 2

    2 2 2

    0 0 0

    4 4 4sin .2cos 4 cos x dx t tdt tdt

    .

    b) t tan , ;2 2

    x t t .

    Khi 0x th 0t , khi 1x th4

    t .

    Ta c:2

    tancos

    dt

    x t dxt

    .

    1 4 4

    2 2 2

    0 0 0

    1. .4

    1 1 tan cos 40

    dx dt dt t

    x t t

    Ch : Trong thc t chng ta c th gp dng tch phn trn dng tng qut hn

    nh:

    Nu hm s di du tch phn c cha cn dng 2 2 2 2,a x a x v

    2 2x a (trong trong a l hng s dng) m khng c cch bin i no

    khc th nn i sang cc hm s lng gic lm mt cn thc, c th l:

    Vi 2 2a x , t sin , ;2 2

    x a t t

    hoc cos , 0;x a t t .

    Vi 2 2a x , t tan , ;2 2

    x a t t

    hoc , 0; x acott t .

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    Vi 2 2x a , t , ; \ 0sin 2 2

    ax t

    t

    hoc ;cos

    ax

    t 0; \

    2t

    .

    *Phng php i bin dng II

    nh l :Nu hm s ( )u u x n iu v c o hm lin tc trn on ;a b

    sao cho '( ) ( ( )) ( ) ( )f x dx g u x u x dx g u du th( )

    ( )

    ( ) ( )

    u bb

    a u a

    I f x dx g u du .

    V d 3:Tnh

    1

    2 3

    05I x x dx

    Gii: t 3( ) 5u x x .Tac (0) 5, (1) 6u u .

    T c: 6

    5

    61 2 2 4 106 6 5 5 6 5

    53 9 9 9 9I udu u u

    V d 4:Hy tnh cc tch phn sau bng phng php i bin dng II:

    a) 1

    5

    0

    2 1x dx b)2

    ln

    e

    e

    dx

    x x c)1

    2

    0

    4 2

    1

    xdx

    x x

    d)

    2

    2

    1(2 1)

    dx

    x e)2

    3

    3

    2cos(3 )

    3x dx

    Gii: a) t 2 1u x khi 0x th 1u . Khi 1x th 3u

    Ta c 22

    dudu dx dx . Do :

    1 3

    65 5 6

    0 1

    31 12 1 (3 1)

    12 12 12

    ux dx u du = 60 23 .

    b)tlnu x

    . Khix e

    th1u

    . Khi 2

    x eth

    2u.

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    Ta cdx

    dux

    2 2

    1

    2ln ln 2 ln1 ln 2

    1ln

    e

    e

    dx duu

    x x u .

    c)t 2 1u x x . Khi 0x th 1u . Khi 1x th 3u .

    Ta c (2 1)du x dx . Do :1 3

    2

    0 1

    34 2 22ln 2(ln3 ln1) 2ln3

    11

    x dudx u

    x x u

    .d)t 2 1u x . Khi 1x th 1u . Khi 2x th 3u .

    Ta c 22

    dudu dx dx . Do :

    2 3

    2 2

    1 1

    31 1 1 1 1( 1)

    1(2 1) 2 2 2 3 3

    dx du

    x u u

    .

    e)t2

    33

    u x

    .

    Khi

    3x

    th

    3u

    ,

    Khi2

    3x

    th

    4

    3u

    .

    Ta c 33

    dudu dx dx . Do :

    2 4

    3 3

    3 3

    4

    2 1 1 1 43cos(3 ) cos sin sin sin3 3 3 3 3 3

    3

    x dx udu u

    1 3 3 3

    3 2 2 3

    .

    2.Phng php tch phn tng phn.

    nh l .Nu u(x) v v(x) l cc hm s c o hm lin tc trn ;a b th:

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    ' '( ) ( ) ( ) ( ) ( ) ( )b b

    a a

    bu x v x dx u x v x v x u x dx

    a

    hay

    b b

    a a

    budv uv vdu

    a

    .

    p dng cng thc trn ta c qui tc cng thc tch phn tng phn sau:

    Bc 1: Vit f(x)dx di dng 'udv uv dx bng cch chn mt phn thchhp ca f(x) lm u(x) v phn cn li ' ( ) .dv v x dx

    Bc 2: Tnh 'du u dx v ' ( )v dv v x dx .

    Bc 3: Tnh 'b b

    a a

    vdu vu dx vb

    uva

    .

    Bc 5: p dng cng thc trn.V d 5:Tnh

    1

    lne

    x xdx

    Gii: tlnu x

    dv xdx

    2

    2

    dxdux

    xv

    2 2 2 2

    1 1

    1 1ln ln

    1 12 2 2 4 4

    e ee ex e x e

    x xdx x xdx

    .

    V d 6:Tnh cc tch phn sau:

    a)

    2

    5

    1

    lnxdx

    x b)2

    0

    cosx xdx

    c)1

    0

    xxe dx d)2

    0

    cosxe xdx

    Gii: a) t

    54

    ln

    1

    14

    dxu x du

    x

    dv dx vx x

    . Do :

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    5 4 5 411 1 1

    ln ln 1 ln 2 1 1 15 4 ln 2

    4 4 64 4 4 256

    x x dxdx

    x x x x

    .

    b) t

    cos sin

    u x du dx

    dv xdx v x

    . Do :

    2 2

    0 0

    cos sin sin cos 12 22 2

    0 0x xdx x x xdx x

    .

    c)tx x

    u x du dx

    dv e dx v e

    . Do :

    1 1

    0 0

    1 11 1

    0 0x x x xxe dx xe e dx e e e e .

    d) t cos sin

    x xu e du e dx

    dv xdx v x

    2 2

    0 0

    cos sin sin20

    x x xe xdx e x e xdx

    .

    t1 1

    1 1sin cos

    x xu e du e dx

    dv xdx v x

    2 2

    2

    0 0cos cos cos2

    0

    x x x

    e xdx e e x e xdx

    .

    2 22

    2

    0 0

    12 cos 1 cos .

    2x x ee xdx e e xdx

    *Cch t u v dv trong phng php tch phn tng phn.

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    ( )b

    x

    a

    P x e dx ( ) lnb

    a

    P x xdx ( )cosb

    a

    P x xdx cosb

    x

    a

    e xdx u P(x) lnx P(x) xe

    dv xe dx P(x)dx cosxdx cosxdx

    Ch : iu quan trng khi s dng cng thc tch phn tng phn l lm th no

    chn u v 'dv v dx thch hp trong biu thc di du tch phn f(x)dx. Ni

    chung nn chn u l phn ca f(x) m khi ly o hm th n gin, chn

    'dv v dx l phn ca f(x)dx l vi phn mt hm s bit hoc c nguyn hm

    d tm.

    C ba dng tch phn thng c p dng tch phn tng phn:

    Nu tnh tch phn ( ) ( )P x Q x dx

    m P(x)l a thc cha x v Q(x) l

    mt trong nhng hm s: , cos , sinaxe ax ax th ta thng t

    ' ( )( )

    ( ) ( )

    du P x dxu P x

    dv Q x dx v Q x dx

    Nu tnh tch phn ( ) ( )P x Q x dx

    m P(x) l a thc ca x v Q(x) l

    hm s ln(ax) th ta t '

    ( )( ) ( )

    du Q x dxu Q xdv P x dx v P x dx

    Nu tnh tch phn cosaxI e bxdx

    hoc sinaxJ e bxdx

    th

    ta t 1cos sin

    axax du ae dxu e

    dv bxdx v bxb

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    hoc t 1sin cos

    axax du ae dxu e

    dv bxdx v bxb

    Trong trng hp ny, ta phi tnh tch phn tng phn hai ln sau tr

    thnh tch phn ban u. T suy ra kt qu tch phn cn tnh.

    II.TCH PHN MT S HM S THNG GP

    1. Tch phn hm s phn thc

    a)Tnh tch phn dng tng qut sau:

    2 0dx

    I aax bx c

    .

    (trong 2 0ax bx c vi mi ;x )

    Xt 2 4b ac .

    +)Nu 0 th 2

    2

    dxI

    ba x

    a

    tnh c.

    +)Nu 0 th 1 2

    1 dxI

    a x x x x

    ,

    (trong 1 2;2 2

    b bx x

    a a

    )

    1

    1 2 2

    1ln

    x xI

    a x x x x

    .

    +) Nu 0 th22 2

    22 4

    dx dx

    Iax bx c b

    a xa a

    t 22 21

    tan 1 tan

    2 4 2

    bx t dx t dt

    a a a

    , ta tnh c I.

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    b) Tnh tch phn: 2 , 0mx n

    I dx aax bx c

    .

    (trong 2

    ( ) mx n

    f x

    ax bx c

    lin tc trn on ; )

    +) Bng phng php ng nht h s, ta tm A v B sao cho:

    cbxax

    B

    cbxax

    baxA

    cbxax

    nmx

    222

    )2(

    +)Ta c I=

    dcbxax

    Bdx

    cbxax

    baxAdx

    cbxax

    nmx

    222

    )2(

    . Tch phn dcbxax

    baxA

    2)2(

    =

    cbxaxA 2ln

    Tch phn2

    dx

    ax bx c

    tnh c.

    c) Tnh tch phn( )

    ( )

    b

    a

    P xI dx

    Q x vi P(x) v Q(x) l a thc ca x.

    Nu bc ca P(x) ln hn hoc bng bc ca Q(x) th dng php chia athc.

    Nu bc ca P(x) nh hn bc ca Q(x) th c th xt cc trng hp:+ Khi Q(x) ch c nghim n

    1 2, ,..., n th t

    1 2

    1 2

    ( )...

    ( )n

    n

    AA AP x

    Q x x x x

    .

    + Khi 2 2( ) , 4 0Q x x x px q p q th t

    2

    ( ).

    ( )

    P x A Bx C

    Q x x x px q

    + Khi 2

    ( )Q x x x vi th t

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    2

    ( )

    ( )

    AP x B C

    Q x x x x

    .

    V d 7. Tnh tch phn:

    1

    2

    0

    4 115 6

    x dxx x

    .

    Gii:

    Cch 1.Bng phng php ng nht h s ta c th tm A, B sao cho:

    2 2 2

    2 54 11, \ 3; 2

    5 6 5 6 5 6

    A xx Bx

    x x x x x x

    2 22 54 11 , \ 3; 2

    5 6 5 6Ax A Bx x

    x x x x

    2 4 2

    5 11 1

    A A

    A B B

    Vy

    2 2 22 2 54 11 1

    , \ 3; 25 6 5 6 5 6

    xxx

    x x x x x x

    .

    Do

    1 1 1

    2 2 2

    0 0 0

    4 11 2 52

    5 6 5 6 5 6

    x x dxdx dx

    x x x x x x

    21 12 9

    2ln 5 6 ln ln0 03 2

    xx x

    x

    .

    Cch 2. V 2 5 6 2 3x x x x nn ta c th tnh tch phn trn bng

    cch:

    Tm A, B sao cho:

    24 11

    , \ 3; 25 6 2 3

    x A Bx

    x x x x

    2 2

    34 11, \ 3; 2

    5 6 5 6

    A B x A Bxx

    x x x x

    4 33 2 11 1A B AA B B

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    Vy 24 11 3 1

    , \ 3; 25 6 2 3

    xx

    x x x x

    .

    Do

    1 1 1

    2

    0 0 0

    4 113

    5 6 2 3

    x dx dxdx

    x x x x

    1 1 93ln 2 ln 3 ln

    0 0 2x x .

    V d 8:Tnh tch phn:

    1

    2

    01

    dx

    x x .Gii:

    Do

    1 1

    22

    0 01 1 3

    2 4

    dx dx

    x xx

    t 21 3 3

    tan , ; 1 tan2 2 6 3 2

    x t t dx t dt

    Vy 21 3 3

    22

    06 6

    3 1 tan 2 3 2 3 33231 3 3 9(1 tan )4 6

    t dtdxdt t

    x x t.

    V d 9. Tnh tch phn:

    12 3

    2

    01

    xdx

    x .Gii:

    1 1 1 1

    2 2 2 23

    2 2 2

    0 0 1 01 1 1

    x x xdxdx x dx xdx

    x x x

    22

    1 11 1 1 3

    ln 1 ln2 2

    2 2 8 2 40 0

    xx .

    2. Tch phn cc hm lng gic

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    http://ebooktoan.com 122.1.Dng 1: Bin i v tch phn c bn

    V d 10:Hy tnh cc tch phn sau:

    a)

    2

    2

    sin2 sin7J x xdx

    ;

    b)

    2

    4 4

    0

    cos (sin cos )K x x x dx

    ;

    c)

    2 3

    0

    4sin

    1 cos

    xM dx

    x

    .

    Gii

    a)

    2 2

    2 2

    1 1cos5 cos9

    2 2J xdx xdx

    1 1 42 2sin5 sin910 18 45

    2 2

    x x

    .

    b) Ta c 24 4 2 2 2 2cos (sin cos ) cos sin cos 2sin cosx x x x x x x x

    21 1 3 1

    cos 1 sin 2 cos 1 1 cos4 cos cos cos42 4 4 4

    x x x x x x x

    3 1

    cos cos5 cos34 8

    x x x .

    2 2 2 2

    4 4

    0 0 0 0

    3 1 1cos (sin cos ) cos cos5 co3

    4 8 8K x x x dx xdx xdx xdx

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    3 1 1 3 1 1 11sin sin5 sin32 2 2

    4 40 24 4 40 24 150 0 0

    x x x

    .

    c)

    3 2 24sin 4sin sin 4(1 cos )sin4(1 cos )sin1 cos 1 cos 1 cos

    x x x x xx xx x x

    2M .

    2.2.Dng 2: i bin s hu t ha tch phn hm lng gic

    2.2.1.Tnhcos

    dxI

    asinx b x c

    Phng php:

    t2

    2tan2 1

    x dtt dxt

    Ta c:2

    2sin

    1

    tx

    t

    v

    2

    2

    1cos

    1

    tx

    t

    22

    cos 2

    dx dt I

    asinx b x c c b t at b c

    bit cch tnh.

    V d 11. Tnh4cos 3sin 5

    dxx x

    Gii: t 22

    1 2tan 1 tan

    2 2 2 1

    x x dtt dt dx dx

    t

    2

    2 2

    2 2

    21

    1 2cos 3sin 3 3 23 31 1

    dtdx dt t

    t tx x t tt t

    tan 11 2ln ln2 tan 2

    2

    xt

    C Cxt

    .

    2.2.2. Tnh 2 2sin sin cos cos

    dx

    I a x b x x c x d

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    Phng php: 2 2sin sin cos cos

    dxI

    a d x b x x c d x

    2

    2

    costan tan

    dxx

    a d x b x c d

    t2cos

    dxt tgx dt

    x

    2dt

    Ia d t bt c d

    tnh c.

    V d 12. Tnh:2 2sin 2sin cos 3cos

    dxI

    x x x x

    .

    Gii:Ta c2

    2 2 2

    cossin 2sin cos 3cos tan 2 tan 3

    dxdx xI

    x x x x x x

    t2

    tancos

    dx

    t x dt x

    21 1 1 tan 1

    ln ln2 3 1 3 4 3 4 tan 3

    dt dt t x

    I C Ct t t t t x

    2.2.3. Tnhsin cossin cos

    m x n x pI dxa x b x c

    .

    Phng php:

    +)Tm A, B, C sao cho:

    sin cos sin cos cos sin ,m x n x p A a x b x c B a x b x C x

    +) Vysin cos

    sin cos

    m x n x pI dx

    a x b x c

    =

    =

    cxbxa

    dxCdx

    cxbxa

    xbxaBdxA

    cossincossin

    sincos

    Tch phn dx tnh c

    Tch phn Ccxbxadx

    cxbxa

    xbxa

    cossinln

    cossin

    sincos

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    Tch phn cxbxadx

    cossin tnh c.

    V d 13. Tnh:cos 2sin

    4cos 3sin

    x xI dx

    x x

    .

    Gii:

    Bng cch cn bng h s bt nh, tm A v B sao cho:

    cos 2sin 4cos 3sin 4sin 3cos , x x A x x B x x x

    cos 2sin 4 3 cos 3 4 sin ,x x A B x A B x x

    2

    4 3 1 53 4 2 1

    5

    A

    A BA B

    B

    2 1 4sin 3cos 2 1. ln 4cos 3sin

    5 5 4cos 3sin 5 5

    x xI dx x x x C

    x x

    .2.3.Dng 3: i bin s a vtch phn hm lng gic n gin hn

    (Xem v d 17, 20, 21)

    2.4.Ch : Nguyn hm dng sin , cosR x x dx , vi sin , cosR x x l mthm hu t theo sinx, cosx

    tnh nguyn hm trn ta i bin s v a v dng tch phn hm hu t

    m ta bit cch tnh tch phn.

    Trng hp chung: t 22

    tan 2 1 x dt

    t dx t

    Ta c2

    2 2

    2 1sin ;cos

    1 1

    t tx x

    t t

    Nhng trng hp c bit:+) Nu sin , cosR x x l mt hm s chn vi sinx v cosx ngha l

    sin , cos sin ,cosR x x R x x th t tant x hoc cott x ,sau a tch phn v dng hu t theo bin t.

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    http://ebooktoan.com 16+) Nu sin , cosR x x l hm s l i vi sinx ngha l:

    sin ,cos sin ,cosR x x R x x th t cost x .

    +) Nu sin , cosR x x l hm s l i vi cosx ngha l:

    sin , cos sin ,cosR x x R x x th t sint x .

    3.Tch phn hm v t

    3.1 .Dng 1: Bin i v tch phn v t c bn

    V d 14. Tnh tch phn:

    1

    01

    dxI

    x x

    .

    Gii

    1 1

    3322

    0 0

    121 1

    031

    dx

    I x x dx x xx x

    2 2 2 23

    V d 15:Tnh tch phn

    1 3

    20 1

    x dx

    x x .

    Gii:

    1 13 3 2 4

    20 0

    2 2 1( 1 )151

    x dx x x x dxx x

    .

    3.2.Dng 2: Bin i v tch phn hm lng gic

    (xem v d 2)

    3.3Dng 3: Bin i lm mt cnGm: i bin s t l ton b cn thc

    Vit biu thc trong cn di dng bnh phng ngV d 15:Tnh

    1

    0

    23 1 dxxxI

    Gii:

    1

    0

    22

    1

    0

    23 .11 xdxxxdxxxI

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    t t= 22222 111 txxtx

    Ta c: xdx=-tdt, Khi x= 0 th t =1,khi x = 1 th t =0

    Vy

    152

    53)1(

    1

    0

    530

    1

    22

    ttdtttI

    4.Tch phn cha du gi tr tuyt i

    Phng php:Chng ta phi ph du gi tr tuyt i

    V d 16:Tnh

    2

    2

    2

    1J x dx

    Gii: Lp bng xt du ca 2 1x trn on 2;2

    x -2 -1 1 2

    2 1x + 0 - 0 +

    Do 2 1 1 2

    2 2 2 2

    2 2 1 1

    1 1 1 1I x dx x dx x dx x dx

    3 3 31 1 2

    42 1 13 3 3

    x x xx x x

    .

    III.TCH PHN MT S HM C BIT

    1.Cho hm s ( )y f x lin tc v l trn on ;a a . Khi

    ( ) 0

    a

    a

    I f x dx

    .

    V d 17:Chng minh

    2

    2

    2

    04 sin

    xdxI

    x

    .

    Gii: t x t dx dt . Khi x= 2

    th t = - 2

    , khi2

    x

    th2

    t

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    Do : I= It

    tdt

    2

    2

    2sin4

    Suy ra : 2I = 0. Ta c2

    2

    2

    04 sin

    xdxIx

    .

    2.Cho hm s ( )y f x lin tc v chn trn on ;a a . Khi

    0

    ( ) 2 ( )a a

    a

    I f x dx f x dx

    .

    Chng minh : Ta c

    0

    0

    ( ) ( ) ( )a a

    a a

    I f x dx f x dx f x dx

    (1)

    Ta tnh

    0

    ( )a

    J f x dx

    bng cch t 0x t t a dx dt 0 0

    0 0

    ( ) ( ) ( ) ( )

    a a

    a a

    J f x dx f t dt f t dt f x dx

    (2)

    Thay (2) vo (1) ta c

    0

    ( ) 2 ( )a a

    a

    I f x dx f x dx

    V d 18:Tnh tch phn:

    2

    2

    2

    cos

    4 sin

    x x

    I dxx

    Gii: Ta c

    2 2 2

    2 2 2

    2 2 2

    cos cos

    4 sin 4 sin 4 sin

    x x x xI dx dx dx

    x x x

    Do 1 2( ) 4 sinxf x

    x

    l hm s l trn ;

    2 2

    nn

    2

    2

    2

    04 sin

    x dxx

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    v 2 2cos

    ( )4 sin

    xf x

    x

    l hm s chn trn ;

    2 2

    nn ta c:

    2 2 2

    2 20

    2 2

    cos cos (sin )2 2

    4 sin 4 sin (sin 2) sin 2

    x x d xdx dx

    x x x x

    Vy1 sin 2 1

    ln ln322 sin 2 2

    0

    xI

    x

    .

    3.Cho hm s ( )y f x lin tc v chn trn on : . Khi

    dxxfdxa

    xfIx

    )(21

    1)(

    Chng minh: t t= -x dt= - dx

    Ta c f(x) = f(-t)= f(t); ax+1= a-t+1=1t

    t

    a

    a

    Khi x= - th t = ; x = th t =-

    Vy

    dttfa

    adt

    a

    tfadx

    a

    xfI

    t

    t

    t

    t

    x)(

    1

    11

    1

    )(

    1

    )(

    Idxxfdta

    tfdttf

    t)(

    1

    )()(

    Suy ra

    dxxfdxa

    xf

    I x )(2

    1

    1

    )(

    V d 19:Tnh tch phn:1

    4

    12 1xx

    I dx

    .

    Gii:t t= -x dt= - dx

    Khi x= - 1 th t = 1 ; x =1 th t =-1

    Vy

    1

    1

    1

    `1

    441

    1

    4

    122

    1212 dttdt

    tdx

    xI

    t

    t

    tx

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    1

    1

    1

    1

    1

    1

    44

    4

    12 Idxxdt

    tdtt

    t

    Suy ra 5

    1

    52

    1

    2

    11

    1

    51

    1

    4

    x

    dxxI

    4.Cho f(x) lin tc trn on 0;2

    .Khi

    2 2

    0 0

    (sin ) (cos )f x dx f x dx

    .Chng minh:

    t2

    t x dx dt

    Khi x = 0 th2

    t , khi

    2x

    th t = 0

    Do

    02 2 2

    0 0 02

    (sin ) (sin( ) (cos ) (cos )2

    f x dx f t dt f t dt f x dx

    .

    Nhn xt: Bng cch lm tng t ta c cc cng thc*Nu f(x) lin tc trn 0;1 th (sin ) (sin )

    2xf x dx f x dx

    *Nu f(x) lin tc trn 0;1 th2 2

    (cos ) (cos )

    xf x dx f x dx

    V d 20:Chng minh: I=

    2

    0

    sin

    sin cos 4

    n

    n n

    xdx

    x x

    .

    .

    Gii :

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    http://ebooktoan.com 21Tng t nh trn ta c:

    I=

    2 2

    0 0

    sin cos

    sin cos sin cos

    n n

    n n n n

    x xdx dx

    x x x x

    =J

    +) Vy I+J=

    2 2

    0 0

    sin cos

    sin cos sin cos 2

    n n

    n n n n

    x xdx dx

    x x x x

    Vy I=

    2

    0

    sin

    sin cos 4

    n

    n n

    xdx

    x x

    .

    V d 21:Tnh tch phn:2

    0

    sin1 cosx x

    dxx

    .

    Gii: t 0x t t dx dt .

    Khi

    0

    2 2

    0

    sinsin

    1 cos 1 cos

    t tx xdx dt

    x t

    2 2

    0 0

    2 2

    0 0

    sin sin

    1 cos 1 cos

    sin sin

    1 cos 1 cos

    t t tdt dt

    t t

    x x xdx dx

    x x

    2 2

    0 0

    sin sin

    2 1 cos 1 cos

    x x x

    dx dxx x

    Vy2

    2 2

    0 0

    sin sin

    1 cos 2 1 cos 4

    x x xdx dx

    x x

    .

    BI TP NGHBi 1.Tnh cc tch phn sau

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    2

    022 sin4cos

    2sin)

    dxxx

    xIa

    ( H-KA-2006)

    2

    0 cos31sin2sin)

    dxxxxIc

    (H-KA-2005)

    2

    0 cos1

    cos.2sin)

    dx

    xxIe

    (H-KB-2005)

    2

    4

    2sin1 cossin)

    dxxxxIg

    2

    03)3cos(sin

    2cos)

    dxxx

    xIi

    2

    0

    sin)

    dxxxIb

    2

    0

    2

    .cos)12()

    dxxxId

    4

    0 2cos1)

    dxx

    xIf

    3

    4

    2cos1cos

    tan)

    dxxx

    xIh

    4

    0

    2 .tan)

    dxxxIk

    Bi 2.Tnh cc tch phn sau

    3

    0

    2

    35

    1

    2) dx

    x

    xxIa

    4

    0 121

    12) dx

    x

    xIc

    dxxxIe 3

    1

    23 1.)

    32

    52 4

    )xx

    dxIg

    3

    1

    22

    )1(

    )

    xx

    dxIb

    1

    2

    12

    11

    1) dx

    xxId

    3

    13

    )xx

    dxIf

    5

    3

    22) dxxxIh

    Bi 3. Tnh cc tch phn sau

    1

    0

    2 )1() dxexIa x

    1

    0 1)

    xe

    dxIc

    2

    02

    2

    )2(

    .

    ) dxx

    ex

    Ie

    x

    2

    12

    )1ln() d

    x

    xIb

    e

    dxxx

    xId

    1

    3

    .ln1

    )

    3

    2

    2

    ).ln() dxxxIf

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    0

    1

    32 )1() dxxexIg x 2

    0

    sin .cos)cos()

    dxxxeIh x