ΑΠΕΙΡΟΣΤΙΚΟΣ ΙΙ

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  • Riemann

    .

    JJ II

    J I

    1 pi 273

    pi url: http://www.aegean.gr

    pi

    pipi

    832 00

    Copyright pi ,

    All rights reserved

  • Riemann

    .

    JJ II

    J I

    2 pi 273

    1

    1.1.

    1.1.1 f : I R . F : I R

    x I, F (x) = f (x), F pi f

    f (x)dx. F (x) =

    f (x)dx.

    . F (x) pi f (x) H(x) = F (x)+c,c . H(x) = F (x) + c f .

    1)

    xndx = x

    n+1

    n+1 + c, n N2)

    dxx = ln(x) + c

    3)xadx = x

    a+1

    a+1 + c, a R

  • Riemann

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    JJ II

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    3 pi 273

    4)

    sin(x)dx = cos(x) + c5)

    cos(x)dx = sin(x) + c

    6)

    dxcos2(x) = tan(x) + c

    7)

    dxsin2(x) = cot(x) + c

    8)axdx = a

    x

    ln(a) + c

    9)exdx = ex + c

    10)

    dx1x2 = arcsin(x) + c

    11)

    dx1+x2

    = arctan(x) + c

    12)

    sinh(x)dx = cosh(x) + c12)

    cosh(x)dx = sinh(x) + c

    1.1.2 f 1, f 2 : I R pi - pi h(x) = c1f1(x) + c2f2(x) pi c1, c2 R

    (c1f1(x) + c2f2(x)) = c1

    f1(x)dx + c2

    f2(x)dx

    1.1.3 1, 2 : 1 2 , f : 2 R - (t) , 0.

    f (x)dx =

    f ((t))(t)dt

    : pi , pi pi, pi pi.

    pi pia2 x2 x = |a| sin(u)

    x = |a| cos(u).

  • Riemann

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    4 pi 273

    pi pia2 + x2 x = |a| tan(u)

    x = |a| cot(u). pi pi

    x2 a2 x = |a| 1cos(u) x =|a| cosh(u).

    pi piax + b t = ax + b.

    pi pi2ax x2 x = a(1 cos(u)).

    1.1.4 f, g : R , pi f , g pi

    f (x)g(x)dx = f (x)g(x)

    f (x)g(x)dx, x

    : I =

    P(x)eaxdx, a R, P(x) pi

    I =

    P(x)eaxdx =

    1a

    P(x)deax

    =1aP(x)eax 1

    a

    P (x)eaxdx.

    I =P(x) cos(ax + b)dx, a, b R, P(x) pi

    I =

    P(x) cos(ax + b)dx =

    1a

    P(x)d sin(ax + b)

    =1aP(x) sin(ax + b) 1

    a

    P (x) sin(ax + b)dx.

    ( cos sin.)

  • Riemann

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    5 pi 273

    I =P(x)ekx cos(ax + b)dx, a, b R, P(x) pi-

    f (x) =ekx sin(ax + b)dx I =

    P(x)df (x) = P(x)f (x) f (x)P (x)dx.

    ( cos sin.)

    I =

    ekx cos(ax + b) sin(ax + b)dx I =

    ekx sin(ax + b) sin(ax + b)dx

    I =ekx cos(ax + b) cos(ax + b)dx, a, b R pi

    2 sin(a) sin(b) = cos(a b) cos(a + b),2 sin(a) cos(b) = sin(a + b) + sin(a b),2 cos(a) cos(b) = cos(a + b) + cos(a b).

    I =f (x) ln((x))dx I =

    f (x) arcsin((x))dx

    I =f (x) arccos((x))dx

    f (x) arctan((x))dx, pi f ,

    F =f

    f (x) ln((x))dx =

    ln((x))df (x) = F (x) ln((x)) F (x)(x)(x) dx.

    I =

    f (x)2(x)dx, pi

    1(x)

    .

    1.1.5 f : R R f (x) = p(x)q(x) pip, q pi.

    : R , R(x) = P(x)Q(x) pi m 0 P(x) n

    Q(x) 0. m n P(x) = P1(x)Q(x) + P2(x) pi piP1(x) P2(x). P(x)Q(x) = P1(x) +

    P2(x)Q(x) , pi deg(P2(x)) n. pi

    pi pipi :

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    6 pi 273

    m n pipi pi Q(x) pi R(x) pi pi pi Q(x). Q(x) pi pi P(x)Q(x) =

    A1(xr1) +

    A2(xr1) + . . .

    An(xrn ) .

    Q(x) pipi pi

    P(x)Q(x)dx =

    (x)Q1(x)

    +1(x)Q2(x)

    dx,pi Q1(x) = (Q(x), Q(x)), Q2(x) = Q(x)Q1(x) deg((x)) deg(Q1(x)) 1,deg(1(x)) deg(Q2(x)) 1. ( (x) 1(x) pi pi-

    P(x)Q(x)dx =

    (x)Q1(x)

    +

    1(x)Q2(x)

    dx.

    pi pi - , .

    :

    R

    (x, n

    ax+bcx+d

    )dx, ( ax+bcx+d > 0 n )

    ax+bcx+d =

    tn.

    R

    (x,ax2 + bx + c

    )dx :

    a > 0 ax2 + bx + c = t ax ax2 + bx + c = t + ax.

    a < 0 = b2 4ac > 0 ax2 + bx + c = t |x r1| pi r1 ax2 + bx + c = 0. a < 0 c > 0

    ax2 + bx + c = tx c ax2 + bx + c = tx + c.

    1.2.

    1.2.1 pi ((x) + 1)(x +

    (x) + 1)dx.

    pi-

    1.2.2 pi x2ex

    3dx. pi-

  • Riemann

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    1.2.3 pi

    1ex+1dx. pi-

    1.2.4 pi

    x+22x1dx. pi-

    1.2.5 pi

    1sin(x)dx. pi-

    1.2.6 pi

    1 x2dx. pi-

    1.2.7 pi

    1(xa)(bx)dx a < b.

    pi-

    1.2.8 pi

    (x a)(b x)dx a < b .pi-

    1.2.9 pi

    ex

    e2x+1dx. pi-

    1.2.10 pi

    13+x2dx. pi-

    1.2.11 pi

    x2 5dx. pi-

    1.2.12 pi

    xxx2dx. pi-

    1.2.13 pi

    cos(ax + b)dx. pi-

    1.2.14 pi

    5x+3x2+2x3dx. pi-

    1.2.15 pi

    11+cos(x)dx. pi-

    1.2.16 pi

    x4+x2

    dx. pi-

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    1.2.17 pi

    1xcos(x)dx. pi-

    1.2.18 pi

    2x2+5x1x3+x22x dx. pi-

    1.2.19 pi

    x2+2x+3(x1)(x+1)2dx. pi-

    1.2.20 pi

    3x2+2x2x31 dx. pi-

    1.2.21 pi

    1(x31)2dx. pi-

    1.2.22 pi x2exdx. pi-

    1.2.23 a b pi pi - C(x) =

    eax cos(bx)dx S(x) =

    eax sin(bx)dx. pi-

    1.2.24 pi x2 cos(x)dx. pi-

    1.2.25 pi

    1x ln(x)dx, x > 1. pi-

    1.2.26 pi

    ln(x)x dx, x > 1. pi-

    1.2.27 pi

    ln(x)dx. pi-

    1.2.28 pi (x2 1) cos(3x)dx. pi-

    1.2.29 pi xex cos(x)dx. pi-

    1.2.30 pi ex sin(2x) cos(x)dx. pi-

  • Riemann

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    1.2.31 pi

    3x212x

    xarctan(x)dx. pi-

    1.2.32 In =

    1(ax2+bx+c)n dx

    In+1 =2ax+b

    n(4acb2)(ax2+bx+c)n +2(2n1)an(4acb2) In. pi-

    1.2.33 pi (

    1+x1x

    ) 12 dx. pi-

    1.2.34 pi

    143xx2dx. pi-

    1.2.35 pi

    1x+

    x2x1dx. pi-

  • Riemann

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    10 pi 273

    2

    Riemann

    2.1.

    2.1.1 I = [a. b] f : I R . P = {x0, x1, x2, . . . , xn} a = x0 < x1 < x2 < . . . 0, pi > 0 |P | < = |L(P, f ) U (P, f )| < 2.1.8 f : [a, b] R Riemann, baf (x)dx, pi.

    Riemann

    2.1.9 f : [a, b] R c R bacf (x)dx =

    c baf (x)dx.

  • Riemann

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    12 pi 273

    2.1.10 f, g : [a, b] R bag(x) + f (x)dx =

    baf (x)dx +

    bag(x)dx.

    2.1.11 f : [a, b] R c R baf (x)dx = c

    af (x)dx +

    bcf (x)dx.

    2.1.12 f : [a, b] R | baf (x)dx | b

    a|f (x)|dx.

    2.1.13 f : [a, b] R F (x) = xaf (t)dt

    2.1.14 f : [a, b] R pi (a, b) baf (x)dx = f ( )(b a).

    2.1.15 f : [a, b] R F (x) = xaf (t)dt

    x [a, b] F (x) = f (x). 2.1.16 f : [a, b] R H : [a, b] R pi f(H = f, x [a, b]), b

    af (t)dt = H(a) H(b).

    pi

    2.1.17 f, g : [a, b] R baf (x)g(x)dx =

    f (x)g(x)|ba baf (x)g(x)dx

    2.1.18 g : [a, b] R pi f :[g(a), g(b)] R g(b)

    g(a) f (x)dx = baf (g(x))g(x)dx.

    .

  • Riemann

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    13 pi 273

    2.1.19 f, g : [a, b] R . g pi pi [a, b] f ( ) b

    ag(x)dx =

    baf (x)g(x)dx.

    2.1.20 f : [a, b] R G(x) = bxf (t)dt pi

    G(x) = f (x) x [a, b]. 2.1.21 f, g : [a, b] R . g pi f pi [a, b] b

    af (x)dx = f (a)

    ag(x)dx +

    f (b) bg(x)dx.

    Bonnet 2.1.22 f, g : [a, b] R . f , g pi pi [a, b] b

    af (x)g(x)dx =

    f (a) ag(x)dx.

    2.1.23 f, g : [a, b] R . f , g pi pi [a, b] b

    af (x)g(x)dx =

    f (b) bg(x)dx.

    b

    af (x)dx ' x ni=1 yi , yi = f (xi) = f (a + i ban ) pi b

    af (x)dx ' x2 (y0 + 2y1 + 2y2 + . . . + 2yn1 + yn). Simpson b

    af (x)dx ' x3 {(y0 + yn) + 4(y1 + y3 + . . . + yn1) + 2(y2 + y4 + . . . + yn)}.

  • Riemann

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    14 pi 273

    pipi f 0, f : [a, b] R A pi pi pi pi f xx x = a, x = b A =

    baf (x)dx.

    f A = ba|f (x)|dx.

    f, g : [a, b] R 0 g(x) f (x) A(R) R = {f, g, x = a, x = b} A = b

    af (x)dx b

    ag(x)dx =

    ba[f (x) g(x)]dx.

    x = g(t), y = f (t), t [t1, t2] g(t) , 0 x = g(t), y = f (t) y x

    2.1.24 g(x) : [a, b] R x = g(t), y = f (t) E =

    bag(x)dx =

    t2t1f (t)g(t)dt, g(t1) = a, g(t2) = b f, g [t1, t2]

    pi

    2.1.25 pi pi x = g(t), y = f (t), t [a, b] g, f [a, b] S = L() =

    ba

    g(t)2 + f (t)2dt.

    pi pi pi

    2.1.26 pi pi x = g(t), y = f (t), t [a, b] g, f [a, b] pi pi pi xx B = 2pi

    ba|f (t)|g(t)2 + f (t2)dt.

    pi y = f (x), x [a, b] B = 2pi ba|f (t)|1 + f (x)2dx

    pi pi f : [a, b] R R = {f, Ox, x = a, x = b} pi pi

  • Riemann

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    15 pi 273

    f pi Ox x = a, x = b, V = pi

    baf (x)2dx

    f, g : [a, b] R 0 g(x) f (x) pi pi pipi f g, R = {f, g, Ox, x = a, x = b} V = pi

    ba{f (x)2 g(x)2}dx.

    x = g(t), y = f (t), t = [t1, t2] V = pi t2t1{f (t)2g(t)}dt g(t1) = a, g(t2) = b.

    2.2.

    2.2.1 pi Riemann

    limn

    1n

    nk=1

    nek

    pi-

    2.2.2 f, g, h : [a, b] R f (t) g(t) h(t), t [a, b]. f, h R([a, b]), pi R([a, b]) pi Riemann [a, b],

    bafdx =

    bahdx g Riemann -

    , g R([a, b]) bagdx =

    bafdx. pi-

    2.2.3 pi f : [a, b] R Riemann pi

    ba|f (t)|dt = 0 f = 0. pi-

    2.2.4 pi n = 1,2, 12 +

    13 + +

    1n a > 0. pi-

    2.2.7 f : [a, b] R pi

    limn

    b an

    nk=1

    f(a + k

    b an

    )=

    baf (x)dx.

    pi-

    2.2.8