Giai Bai Tap Tich Phan Suy Rong p2

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  • GII BI TP TCH PHN SUY RNG

    Bi ging in t

    TS. L Xun i

    Trng i hc Bch Khoa TP HCM

    Khoa Khoa hc ng dng, b mn Ton ng dng

    Email: ytkadai@hcmut.edu.vn

    TP. HCM 2015.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 1 / 36

  • Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

    Tnh tch phn suy rng loi 1

    Cu 1

    Tnh tch phn suy rng I =

    +1

    dx

    xx2 + x + 1

    .

    t t =1

    x x = 1

    t dx = dt

    t2.

    i cn

    x 1 +t 1 0

    .

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 2 / 36

  • Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

    Tnh tch phn suy rng loi 1

    Cu 1

    Tnh tch phn suy rng I =

    +1

    dx

    xx2 + x + 1

    .

    t t =1

    x x = 1

    t dx = dt

    t2.

    i cn

    x 1 +t 1 0

    .

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 2 / 36

  • Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

    Khi

    I =

    01

    dtt2

    1t .

    1t2+ 1t + 1

    =

    10

    dt1 + t + t2

    =

    10

    d(t + 12

    )(t + 12

    )2+ 34

    =

    [ln

    t + 12 +t2 + t + 1]1

    0

    = ln

    (3

    2+3

    ) ln

    (3

    2

    )= ln

    (1 +

    23

    ).

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 3 / 36

  • Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

    Tnh tch phn suy rng loi 1

    Cu 2

    Tnh tch phn suy rng I =

    +1

    arctan x

    x2dx .

    t

    u = arctan xdv = dxx2

    du =

    dx

    1 + x2

    v = 1x

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 4 / 36

  • Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

    Tnh tch phn suy rng loi 1

    Cu 2

    Tnh tch phn suy rng I =

    +1

    arctan x

    x2dx .

    t

    u = arctan xdv = dxx2

    du =

    dx

    1 + x2

    v = 1x

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 4 / 36

  • Tch phn suy rng loi 1 Tnh tch phn suy rng loi 1

    Khi

    I =

    [1x. arctan x

    ]+1

    +

    +1

    dx

    x(1 + x2)=

    =pi

    4+

    +1

    dx

    x

    +1

    xdx

    1 + x2=pi

    4+

    [ln |x | 1

    2ln(1 + x2)

    ]+1

    =

    =pi

    4+

    [ln

    x1 + x2]+

    1

    =pi

    4+ ln 1 ln 1

    2=pi

    4+

    ln 2

    2

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 5 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    ngha hnh hc

    Trong trng hp f (x) > 0,x [a,+), gi trca tch phn suy rng hi t c ngha hnh hc

    l din tch ca hnh phng v hn c gi hn

    bi x = a, trc Ox v th hm f (x)

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 6 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Ch .

    T ngha hnh hc ca tch phn suy rng, ta

    c nu tn ti gii hn hu hn v khc 0

    limx+ f (x) = A 6= 0

    v f (x) kh tch trn mi on [a, b] [a,+)th tch phn suy rng

    +a

    f (x)dx phn k

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 7 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    1

    Nu > 1 th I =+a

    dx

    xhi t.

    2

    Nu 6 1 th I =+a

    dx

    xphn k.

    1

    Nu > 1 th I =+2

    dx

    x. ln xhi t.

    2

    Nu < 1 th I =+2

    dx

    x. ln xphn k.

    3

    Nu = 1 th I =+2

    dx

    x. ln xhi t nu

    > 1, phn k nu 6 1.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    1

    Nu > 1 th I =+a

    dx

    xhi t.

    2

    Nu 6 1 th I =+a

    dx

    xphn k.

    1

    Nu > 1 th I =+2

    dx

    x. ln xhi t.

    2

    Nu < 1 th I =+2

    dx

    x. ln xphn k.

    3

    Nu = 1 th I =+2

    dx

    x. ln xhi t nu

    > 1, phn k nu 6 1.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    1

    Nu > 1 th I =+a

    dx

    xhi t.

    2

    Nu 6 1 th I =+a

    dx

    xphn k.

    1

    Nu > 1 th I =+2

    dx

    x. ln xhi t.

    2

    Nu < 1 th I =+2

    dx

    x. ln xphn k.

    3

    Nu = 1 th I =+2

    dx

    x. ln xhi t nu

    > 1, phn k nu 6 1.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    1

    Nu > 1 th I =+a

    dx

    xhi t.

    2

    Nu 6 1 th I =+a

    dx

    xphn k.

    1

    Nu > 1 th I =+2

    dx

    x. ln xhi t.

    2

    Nu < 1 th I =+2

    dx

    x. ln xphn k.

    3

    Nu = 1 th I =+2

    dx

    x. ln xhi t nu

    > 1, phn k nu 6 1.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    1

    Nu > 1 th I =+a

    dx

    xhi t.

    2

    Nu 6 1 th I =+a

    dx

    xphn k.

    1

    Nu > 1 th I =+2

    dx

    x. ln xhi t.

    2

    Nu < 1 th I =+2

    dx

    x. ln xphn k.

    3

    Nu = 1 th I =+2

    dx

    x. ln xhi t nu

    > 1, phn k nu 6 1.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 8 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Cu 3

    Tm tch phn sau hi t I =

    +1

    dx

    x 31 + x2

    1

    x 31 + x2

    x+ 1x.x2/3

    =1

    x+2/3

    tch phn I hi t th +2

    3> 1 > 1

    3.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 9 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Cu 3

    Tm tch phn sau hi t I =

    +1

    dx

    x 31 + x2

    1

    x 31 + x2

    x+ 1x.x2/3

    =1

    x+2/3

    tch phn I hi t th +2

    3> 1 > 1

    3.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 9 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Cu 4

    Tm tch phn sau hi t

    I =

    +1

    (2x + 3)dx

    (4 + x) 31 + x4

    Trng hp 1: > 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2xx.x4/3

    =2

    x+1/3

    tch phn I hi t th +1

    3> 1 > 2

    3.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 10 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Cu 4

    Tm tch phn sau hi t

    I =

    +1

    (2x + 3)dx

    (4 + x) 31 + x4

    Trng hp 1: > 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2xx.x4/3

    =2

    x+1/3

    tch phn I hi t th +1

    3> 1 > 2

    3.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 10 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Trng hp 2: = 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2x5.x4/3

    =2

    5x1/3

    I phn k v1

    3< 1

    Trng hp 3: < 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2x4.x4/3

    =1

    2x1/3

    I phn k v1

    3< 1.

    Vy tch phn I hi t th > 2/3.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Trng hp 2: = 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2x5.x4/3

    =2

    5x1/3

    I phn k v1

    3< 1

    Trng hp 3: < 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2x4.x4/3

    =1

    2x1/3

    I phn k v1

    3< 1.

    Vy tch phn I hi t th > 2/3.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Trng hp 2: = 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2x5.x4/3

    =2

    5x1/3

    I phn k v1

    3< 1

    Trng hp 3: < 0

    (2x + 3)

    (4 + x) 31 + x4

    x+ 2x4.x4/3

    =1

    2x1/3

    I phn k v1

    3< 1.

    Vy tch phn I hi t th > 2/3.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 11 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Cu 5

    Tm tch phn sau hi t

    I =

    +1

    (3x + 4x)dx(5 + x)1

    Trng hp 1: > 0

    (3x + 4x)(5 + x)1

    x+ 4xx(1)

    =4

    x(1)1

    tch phn I hi t th

    ( 1) 1 > 1 > 2 < 1 > 2.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 12 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Cu 5

    Tm tch phn sau hi t

    I =

    +1

    (3x + 4x)dx(5 + x)1

    Trng hp 1: > 0

    (3x + 4x)(5 + x)1

    x+ 4xx(1)

    =4

    x(1)1

    tch phn I hi t th

    ( 1) 1 > 1 > 2 < 1 > 2.TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 12 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Trng hp 2: = 0

    (3x + 4x)(5 + x)1

    x+ 4x61

    =4

    61.x1

    I phn k v 1 < 1

    Trng hp 3: < 0

    (3x + 4x)(5 + x)1

    x+ 4x51

    =4

    51.x1

    I phn k v 1 < 1Vy tch phn I hi t th > 2.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY RNG TP. HCM 2015. 13 / 36

  • Tch phn suy rng loi 1 Tm tch phn suy rng loi 1 hi t

    Trng hp 2: = 0

    (3x + 4x)(5 + x)1

    x+ 4x61

    =4

    61.x1

    I phn k v 1 < 1Trng hp 3: < 0

    (3x + 4x)(5 + x)1

    x+ 4x51

    =4

    51.x1

    I phn k v 1 < 1

    Vy tch phn I hi t th > 2.

    TS. L Xun i (BK TPHCM) GII BI TP TCH PHN SUY