ΑΠΕΙΡΟΣΤΙΚΟΣ ΙΙ

of 273/273
Αόριστο Ολοκλήρωμα Ασκήσεις Ολοκλήρωμα Riemann Ασκήσεις Σειρές Ασκήσεις Γενικευμένα ολοκληρώματα Ασκήσεις Τμ. Μαθηματικών Πρώτη Σελίδα Σελίδα 1 από 273 Πίσω Όλη η οθόνη Κλείσε Έξοδος Πανεπιστήμιο Αιγαίου url: http://www.aegean.gr Ασκήσεις στον Απειροστικό Λογισμό ΙΙ Χρήστος Νικολόπουλος Πανεπιστήμιο Αιγαίου Τμήμα Μαθηματικών 832 00 Καρλόβασι Σάμος © Copyright Πανεπιστήμιο Αιγαίου, Τμήμα Μαθηματικών All rights reserved
  • date post

    22-Nov-2015
  • Category

    Documents

  • view

    29
  • download

    1

Embed Size (px)

Transcript of ΑΠΕΙΡΟΣΤΙΚΟΣ ΙΙ

  • 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

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    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 :

  • Riemann

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    7 pi 273

    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-

  • Riemann

    .

    JJ II

    J I

    8 pi 273

    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

    .

    JJ II

    J I

    9 pi 273

    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

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    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

    .

    JJ II

    J I

    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 f : [a, b] R limn 1nn

    i=1 f (in )

    . pi-

    2.2.9 pi

    an =1

    n + 1 +1

    n + 2 + +1

    n + n

    pi-

    2.2.10 pi 10 e

    xdx. pi-

    2.2.11 f [a, b] pi c [a, b] f (c) > 0,

    baf (x)dx > 0. pi-

  • Riemann

    .

    JJ II

    J I

    17 pi 273

    2.2.12 pi )

    10 x

    2dx

    ) 10 x

    3dx

    ) pi0 cos(x)dx

    ) 11(2x

    2 x3)dx) 32 e

    x/2dxpi-

    2.2.13 pi pi

    pi0 (x + pi) sin(x)dx.

    pi-

    2.2.14 ba

    sin(x)x dx, 0 < a < b 0

    pi-

    2.2.15 pi pi - f (x) = x3 x2 x + 1, g(x) = x + 1. pi- 2.2.16 pi pi pi y =

    1 sin(x) [0,2pi]. pi-

    2.2.17 pi pi piy = x2, x + y = 2. pi-

    2.2.18 pi pi r =6 sin() r = 2(1 cos()) pi- 2.2.19 pi pi y = x2, 0 x 1.pi-

  • Riemann

    .

    JJ II

    J I

    18 pi 273

    2.2.20 pi pix = cos(t)(1 + cos(t)), y = sin(t)(1 + cos(t)), t [0,2pi]. pi-

    2.2.21 pi 41 (2x

    3 5x)dx pi Riemann pi-

    2.2.22 pi 40

    3x + 4dx. pi-

    2.2.23 pi pi pi pi y =sin(x), y = cos(x), x = 0,x = pi/2 pi-

    2.2.24 pi pi a, b, b pi pi pi pi x = a cos(t),y = b sin(t), t [0, pi], (a > b) pi x-. pi-

    2.2.25 pi pi pi pi pi x = a(t sin(t)), y = a(1 cos(t)), t [0,2pi] pi x-.pi-

    2.2.26 pi pi pi pi pi pi x = t2, y = t3 (t

    2 3), pi x-. pi-

    2.2.27 pi pi pi pi pi ,pi pi pi y = x2 y = x + 2, pi x - .pi-

    2.2.28 pi pi pi pi pi ,pi pi x

    2

    a2 +y2

    a2 = 1 pi x - . pi-

  • Riemann

    .

    JJ II

    J I

    19 pi 273

    2.2.29 pi pi pi pi pi -, x = a cos3(t), y = a sin3(t) pi x - . pi-

    2.2.30 pi pi 10

    1 + x4dx -

    pi pi-

    2.2.31 pi pi 10

    1 + x4dx -

    Simpson pi-

  • Riemann

    .

    JJ II

    J I

    20 pi 273

    3

    3.1.

    (a)N pi . G = i=1 ai (G)N .

    3.1.1 (a)N R (G)N . (a, G)

    =1 a.

    3.1.2

    =1 a l R,

    =1 a = l lim G = l l = + pi . l = pi . pi pi.

  • Riemann

    .

    JJ II

    J I

    21 pi 273

    3.1.3

    =1 a (G)N (a)N .

    3.1.4

    =1 a = l

    =1 b = m, l, m R

    =1 a +nb = l +nm.

    =1 a pi pi k pi

    =1 a a > 0, N .

    3.1.5 (G)N .

    3.1.6

    =1 a f : [1, +) R f () = a, N =1 a 1 f (x)dx .

    .

    3.1.7

    =1 a,

    =1 b a b, N1)

    =1 a +

    =1 a

    2)

    =1 a +

    =1 b .

    3.1.8

    =1 a, l = lima+1a

    1) l < 1 =1 a 2) l > 1 =1 a pi .

    . pi .

  • Riemann

    .

    JJ II

    J I

    22 pi 273

    3.1.9

    =1 a .

    3.1.10

    =1 a, (a)N lim a = 0.

    3.1.11

    =1 a pi

    =1 |a | . 3.1.12 pi .

    3.1.13 ( - )

    =1 a 0 N : a q 0, pi a q > 1, > 0.

    3.2.

    3.2.1 n=1 n2n (n+3) .pi-

    3.2.2 pi n=1 3nn!pi-

    3.2.3 n=1 1n pi.pi-

    3.2.4 pi p Z x R n=1 npxnn! .pi-

    3.2.5 n=1 n+2n3 , pi .pi-

  • Riemann

    .

    JJ II

    J I

    23 pi 273

    3.2.6 pi n=1 3nn10 .pi-

    3.2.7 pi n=1 n5n/2 .pi-

    3.2.8 pi n=1 9nn! .pi-

    3.2.9 pi n=1 nn(3n+1)n .pi-

    3.2.10 pi n=1 1+cos2(nx)2n .pi-

    3.2.11 pi limn (n!)n

    nn2= 0.

    pi-

    3.2.12 pi n=1 11+2n1 .pi-

    3.2.13 pi n=1 2n1n .pi-

    3.2.14 pi n=1 12n1 .pi-

    3.2.15 n=1 1na a > 1.pi-

  • Riemann

    .

    JJ II

    J I

    24 pi 273

    3.2.16 pi n=1 n2n .pi-

    3.2.17 pi n=1 nnn! .pi-

    3.2.18 pi n=1 (1 + 1n )n2 .pi-

    3.2.19 pi n=1 ( 12n + 13n ) - pi .

    pi-

    3.2.20 limn n1357(2n1) = 0pi-

    3.2.21 (an)nN 0 an 9, n N. n=1 an10n a 0 a 1.

    pi-

    3.2.22 pi n=1 5n7nn 32 .pi-

    3.2.23 pi n=1 sin( 1n ).pi-

    3.2.24 n=1 1n(ln(n))a a > 1.pi-

    3.2.25 pi pi pi n=1(1)n n2+12n3+n1 .pi-

  • Riemann

    .

    JJ II

    J I

    25 pi 273

    3.2.26 pi pi pi n=1 (1)n+12n .pi-

    3.2.27 pi n=1 2+(1)n2n .pi-

    3.2.28

    1 ln(21 ) +12 ln(

    32 ) + +

    1n ln(n + 1

    n)

    .pi-

  • Riemann

    .

    JJ II

    J I

    26 pi 273

    4

    4.1.

    4.1.1 f : [a,+) R . f pi a +, a

    f (x)dx = limx xaf (t)dt.

    limx xaf (t)dt pi f .

    limx xaf (t)dt + () f pi

    (). pi .

    b f (x)dx = limx

    bxf (t)dt

    f (x)dx = limx

    xcf (t)dt +

    limx cxf (t)dt

  • Riemann

    .

    JJ II

    J I

    27 pi 273

    4.1.2 a

    f (t)dt+ a

    g(t)dt pi a

    lf (t)+mg(t)dt = l a

    f (t)dt+m

    a

    g(t)dt.

    . 4.1.3 f : [a,+) R

    af (x)dx pi

    > 0, pi > 0 y1, y2 [a,+), x1, x2 > ,| x2

    x1f (x)dx | 0, pi > 0 q1, q2 [a, b], |x1 b| < , |x2 b| < | x2

    x1f (x)dx | < .

    4.1.12 f : [a,+b) R+ F (x) = xa|f (t)|dt

    baf (x)dx .

    4.1.13 f, g : [a,+b) R 0 f (x) g(x),a x < b 1)

    bag(x)dx < b

    af (x)dx < .

    2) bag(x)dx = b

    ag(x)dx =

  • Riemann

    .

    JJ II

    J I

    29 pi 273

    4.1.14 f, g : [a,+) R f (x) 0,g(x) 0 x [a,+b) limx f (x)g(x) = l 1) 0 < l < b

    af (x)dx < , b

    ag(x)dx <

    2) l = 0 bag(x)dx < , b

    af (x)dx <

    3) l = baf (x)dx = , b

    ag(x)dx = .

    4.1.15

    n=0 anxn an pi x.

    n=0 an(x a)n an pi x a

    4.1.16 I x I .

    .

    4.1.17 Sn(x) =n1

    m=0 amxm -

    Rn(x) =

    m=n amxm pipi .

    n=0

    anxn = Sn(x) + Rn(x).

    4.1.18 x = x0

    n=0 anxn

    S(x0) = limn Sn(x).

    n=0 anx

    n x = x0 > 0, > 0 : n n0 pin0 Rn(x0) < 0.

  • Riemann

    .

    JJ II

    J I

    30 pi 273

    4.2.

    4.2.1 I = a

    dxxk , a > 0 k > 1

    pi k 1.pi-

    4.2.2 pi 0

    11+x2dx.

    pi-

    4.2.3 pi

    11+x2dx.

    pi-

    4.2.4 pi )

    1

    x1+x2

    ) 0 cos(x)dx

    ) 0 e

    xdx)

    1

    ln(x)x

    pi-

    4.2.5 0 e

    x2dx .pi-

    4.2.6 1

    sin(x)xa dx

    1

    cos(x)xa dx pi-

    a > 1 a > 0.pi-

    4.2.7 pi 3

    dxx2+x2 .

    pi-

  • Riemann

    .

    JJ II

    J I

    31 pi 273

    4.2.8 pi 1

    x

    (1+x2)dx =12 +

    pi2 .

    pi-

    4.2.9 a

    sin(x)x dx pi -

    Cauchypi-

    4.2.10 a

    11+x4dx, a > 0 .

    pi-

    4.2.11 0

    1(1+x3)1/3dx, .

    pi-

    4.2.12 0

    x2

    2x4x2+1dx, .pi-

    4.2.13 0 x sin(x

    4)dx, .pi-

    4.2.14 ba

    dx(xa)k k < 1 pi-

    k > 1.pi-

    4.2.15 ba

    dx(xa)(bx) .

    pi-

    4.2.16 10 sin(

    1x )dx .

    pi-

    4.2.17 pi p pi0

    dxsinp(x) .

    pi-

  • Riemann

    .

    JJ II

    J I

    32 pi 273

    4.2.18 pi

    (x) = +0

    tx1etdt,

    x > 0. ( )pi-

    4.2.19 pi

    B(x, y) = 10tx1(1 t)y1dt,

    x > 0, y < 1. ( )pi-

    4.2.20 pi 0

    cos(x)x

    dx,

    .pi-

    4.2.21 pi 10

    11x2dx .

    pi-

    4.2.22 n=1 n+1n2+1 (x2)n.

    pi-

  • Riemann

    .

    JJ II

    J I

    33 pi 273

    4.2.23 n=1 3n2n+4 xn.pi-

    4.2.24 n=1 xnn! .pi-

    4.2.25 n=1 nn(x1)n.

    pi-

    4.2.26 n=1(1)n+1 xnn .pi-

    4.2.27 pi n=1 xn ln(x)n .pi-

    4.2.28 limn nan = a,

    n=1 anxn R = 1a .

    pi-

    4.2.29 an .

    n=0 an

    2n=0

    an +n=1

    nanxn = 0,

    an.pi-

    4.2.30 pi n=0 anxn R pi n=0 anx2n

    pi-

  • Riemann

    .

    JJ II

    J I

    34 pi 273

    4.2.31 pi n=1 naxn.pi-

  • Riemann

    .

    JJ II

    J I

    35 pi 273

    pi : pi pi .

    1.2.1

  • Riemann

    .

    JJ II

    J I

    36 pi 273

    pi : pi -

    (x)e(x)dx

    1.2.2

  • Riemann

    .

    JJ II

    J I

    37 pi 273

    pi : 1ex+1 = 1 ex

    1+ex .

    1.2.3

  • Riemann

    .

    JJ II

    J I

    38 pi 273

    pi : a b x+22x1 = a +b

    2x1 .

    1.2.4

  • Riemann

    .

    JJ II

    J I

    39 pi 273

    pi : pi sin(x) = 2 sin( x2 ) cos(x2 ).

    1.2.5

  • Riemann

    .

    JJ II

    J I

    40 pi 273

    pi : x = sin(y).

    1.2.6

  • Riemann

    .

    JJ II

    J I

    41 pi 273

    pi : x = a + (b a) sin2(u).

    1.2.7

  • Riemann

    .

    JJ II

    J I

    42 pi 273

    pi : x = a + (b a) sin2(u).

    1.2.8

  • Riemann

    .

    JJ II

    J I

    43 pi 273

    pi : ex = y.

    1.2.9

  • Riemann

    .

    JJ II

    J I

    44 pi 273

    pi : x =3 tan(u).

    1.2.10

  • Riemann

    .

    JJ II

    J I

    45 pi 273

    pi : x =5 cosh(u).

    1.2.11

  • Riemann

    .

    JJ II

    J I

    46 pi 273

    pi : x = 12 (1 cos(u)).

    1.2.12

  • Riemann

    .

    JJ II

    J I

    47 pi 273

    pi : t = ax + b.

    1.2.13

  • Riemann

    .

    JJ II

    J I

    48 pi 273

    pi : a b 5x+3x2+2x3 =a

    x1 +b

    x+3 pi pi pipi.

    1.2.14

  • Riemann

    .

    JJ II

    J I

    49 pi 273

    pi : x = 2 arctan(t).

    1.2.15

  • Riemann

    .

    JJ II

    J I

    50 pi 273

    pi : pi x = 2 tan(u).

    1.2.16

  • Riemann

    .

    JJ II

    J I

    51 pi 273

    pi : pi x = t.

    1.2.17

  • Riemann

    .

    JJ II

    J I

    52 pi 273

    pi : pi pi pi a, b, c 2x2+5x1x3+x22x =

    ax +

    bx1 +

    cx+2

    1.2.18

  • Riemann

    .

    JJ II

    J I

    53 pi 273

    pi : pi pi pi a, b, c x2+2x+3

    (x1)(x+1)2 =a

    x1 +b

    x+1 +c

    (x+1)2 .

    1.2.19

  • Riemann

    .

    JJ II

    J I

    54 pi 273

    pi : pi pi pi a, b, c 3x2+2x2

    x31 =3x2+2x2

    (x1)(x2+x+1) =a

    x1 +bx+c

    x2+x+1 .

    1.2.20

  • Riemann

    .

    JJ II

    J I

    55 pi 273

    pi : pi

    P(x)Q(x)dx =

    (x)Q1(x)

    +

    1(x)Q2(x)

    dx (x),1(x), Q1(x), Q2(x).

    1.2.21

  • Riemann

    .

    JJ II

    J I

    56 pi 273

    pi : pi .

    1.2.22

  • Riemann

    .

    JJ II

    J I

    57 pi 273

    pi : pi - S(x), C(x).

    1.2.23

  • Riemann

    .

    JJ II

    J I

    58 pi 273

    pi : pi pi .

    1.2.24

  • Riemann

    .

    JJ II

    J I

    59 pi 273

    pi : pi pi .

    1.2.25

  • Riemann

    .

    JJ II

    J I

    60 pi 273

    pi : pi pi .

    1.2.26

  • Riemann

    .

    JJ II

    J I

    61 pi 273

    pi : pi pi 1 = (x).

    1.2.27

  • Riemann

    .

    JJ II

    J I

    62 pi 273

    pi : pi pi .

    1.2.28

  • Riemann

    .

    JJ II

    J I

    63 pi 273

    pi : f (x) =ex cos(x)dx pi pi .

    1.2.29

  • Riemann

    .

    JJ II

    J I

    64 pi 273

    pi : pi sin(2x) cos(x) = 12 (sin(3x) + sin(x)), pi pi pi pipi.

    1.2.30

  • Riemann

    .

    JJ II

    J I

    65 pi 273

    pi : pi

    3x212x

    xdx pi pi-

    .

    1.2.31

  • Riemann

    .

    JJ II

    J I

    66 pi 273

    pi : pi .

    1.2.32

  • Riemann

    .

    JJ II

    J I

    67 pi 273

    pi : pi (1+x) pi :

    (1+x1x

    ) 12=

    1+x(1x)(1x) =

    1(1x2) +

    x(1x2) pi pi pipi -

    .

    1.2.33

  • Riemann

    .

    JJ II

    J I

    68 pi 273

    pi : 4 3x x2 = t(x + 4).

    1.2.34

  • Riemann

    .

    JJ II

    J I

    69 pi 273

    pi : x2 x 1 = t x.

    1.2.35

  • Riemann

    .

    JJ II

    J I

    70 pi 273

    pi : pi pi pi . pi Pn pi.. U (f, Pn) .

    2.2.1

  • Riemann

    .

    JJ II

    J I

    71 pi 273

    pi : Pn limn[U (f, Pn) L(g, Pn)] = 0.

    2.2.2

  • Riemann

    .

    JJ II

    J I

    72 pi 273

    pi : f , 0 ba|f |dt pi .

    2.2.3

  • Riemann

    .

    JJ II

    J I

    73 pi 273

    pi : Pn pi.. U ( 1x , Pn) = 1 +12 + 1n1 .

    2.2.4

  • Riemann

    .

    JJ II

    J I

    74 pi 273

    pi : Riemann pi .

    2.2.5

  • Riemann

    .

    JJ II

    J I

    75 pi 273

    pi : Riemann pi .

    2.2.6

  • Riemann

    .

    JJ II

    J I

    76 pi 273

    pi : Riemann.

    2.2.7

  • Riemann

    .

    JJ II

    J I

    77 pi 273

    pi : pi limn bann

    k=1 f(a + k ban

    )=

    baf (x)dx. -

    a b.

    2.2.8

  • Riemann

    .

    JJ II

    J I

    78 pi 273

    pi : pi limn bann

    k=1 f(a + k ban

    )=

    baf (x)dx.

    2.2.9

  • Riemann

    .

    JJ II

    J I

    79 pi 273

    pi : Riemann.

    2.2.10

  • Riemann

    .

    JJ II

    J I

    80 pi 273

    pi : pi f f (x) > 0 pi c.

    2.2.11

  • Riemann

    .

    JJ II

    J I

    81 pi 273

    pi : pi pi .

    2.2.12

  • Riemann

    .

    JJ II

    J I

    82 pi 273

    pi : f (x) = x +pi, g(x) = sin(x) pipi f g.

    2.2.13

  • Riemann

    .

    JJ II

    J I

    83 pi 273

    pi :

    2.2.14

  • Riemann

    .

    JJ II

    J I

    84 pi 273

    pi : f (x) g(x) pi pi f (x) g(x) .

    2.2.15

  • Riemann

    .

    JJ II

    J I

    85 pi 273

    pi : pi 2pi0

    1 sin(x)dx.

    2.2.16

  • Riemann

    .

    JJ II

    J I

    86 pi 273

    pi : pi y = x2, x + y = 2

    2.2.17

  • Riemann

    .

    JJ II

    J I

    87 pi 273

    pi : pi.

    2.2.18

  • Riemann

    .

    JJ II

    J I

    88 pi 273

    pi : pi pi pi -

    ba

    x (t)2 + y(t)2dt.

    2.2.19

  • Riemann

    .

    JJ II

    J I

    89 pi 273

    pi : pi ba

    x (t)2 + y(t)2dt.

    2.2.20

  • Riemann

    .

    JJ II

    J I

    90 pi 273

    pi : pi limn bann

    k=1 f(a + k ban

    )=

    baf (x)dx.

    2.2.21

  • Riemann

    .

    JJ II

    J I

    91 pi 273

    pi : u = 3x + 4.

    2.2.22

  • Riemann

    .

    JJ II

    J I

    92 pi 273

    pi : pi pi pi pi pi x = 0 x = pi/2.

    2.2.23

  • Riemann

    .

    JJ II

    J I

    93 pi 273

    pi : pi pi pi pi x - pi E = 2pi

    ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t),

    l t m.

    2.2.24

  • Riemann

    .

    JJ II

    J I

    94 pi 273

    pi : pi pi pi pi x - pi E = 2pi

    ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t),

    l t m.

    2.2.25

  • Riemann

    .

    JJ II

    J I

    95 pi 273

    pi : pi pi pi pi x - pi E = 2pi

    ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t),

    l t m.

    2.2.26

  • Riemann

    .

    JJ II

    J I

    96 pi 273

    pi : pi pi pi f , g, f (x) g(x) 0, x [a, b] V = pi b

    a{f (x2) g(x)2}dx.

    2.2.27

  • Riemann

    .

    JJ II

    J I

    97 pi 273

    pi : pi pi pi f , x [a, b] V = pi b

    af (x)2dx.

    2.2.28

  • Riemann

    .

    JJ II

    J I

    98 pi 273

    pi : pi pi pi - x = g(t), y = f (t), t [t1, t2] V = pi

    t2t1f (t)2g(t)dt.

    2.2.29

  • Riemann

    .

    JJ II

    J I

    99 pi 273

    pi : [0,1] 10 pi - pi.

    2.2.30

  • Riemann

    .

    JJ II

    J I

    100 pi 273

    pi : [0,1] 10 pi - Simpson.

    2.2.31

  • Riemann

    .

    JJ II

    J I

    101 pi 273

    pi : pi .

    3.2.1

  • Riemann

    .

    JJ II

    J I

    102 pi 273

    pi : pi .

    3.2.2

  • Riemann

    .

    JJ II

    J I

    103 pi 273

    pi : pi .

    3.2.3

  • Riemann

    .

    JJ II

    J I

    104 pi 273

    pi : pi .

    3.2.4

  • Riemann

    .

    JJ II

    J I

    105 pi 273

    pi : pi n=1 1nk .

    3.2.5

  • Riemann

    .

    JJ II

    J I

    106 pi 273

    pi : pi .

    3.2.6

  • Riemann

    .

    JJ II

    J I

    107 pi 273

    pi : pi .

    3.2.7

  • Riemann

    .

    JJ II

    J I

    108 pi 273

    pi : pi .

    3.2.8

  • Riemann

    .

    JJ II

    J I

    109 pi 273

    pi : pi .

    3.2.9

  • Riemann

    .

    JJ II

    J I

    110 pi 273

    pi : pi .

    3.2.10

  • Riemann

    .

    JJ II

    J I

    111 pi 273

    pi : n=1 (n!)nnn2 .

    3.2.11

  • Riemann

    .

    JJ II

    J I

    112 pi 273

    pi : pi .

    3.2.12

  • Riemann

    .

    JJ II

    J I

    113 pi 273

    pi : pi .

    3.2.13

  • Riemann

    .

    JJ II

    J I

    114 pi 273

    pi : pi .

    3.2.14

  • Riemann

    .

    JJ II

    J I

    115 pi 273

    pi : pi .

    3.2.15

  • Riemann

    .

    JJ II

    J I

    116 pi 273

    pi : pi .

    3.2.16

  • Riemann

    .

    JJ II

    J I

    117 pi 273

    pi : pi .

    3.2.17

  • Riemann

    .

    JJ II

    J I

    118 pi 273

    pi : pi .

    3.2.18

  • Riemann

    .

    JJ II

    J I

    119 pi 273

    pi : pi .

    3.2.19

  • Riemann

    .

    JJ II

    J I

    120 pi 273

    pi : n=1 n1357(2n1) .

    3.2.20

  • Riemann

    .

    JJ II

    J I

    121 pi 273

    pi : pi .

    3.2.21

  • Riemann

    .

    JJ II

    J I

    122 pi 273

    pi : pi .

    3.2.22

  • Riemann

    .

    JJ II

    J I

    123 pi 273

    pi : pi .

    3.2.23

  • Riemann

    .

    JJ II

    J I

    124 pi 273

    pi : pi .

    3.2.24

  • Riemann

    .

    JJ II

    J I

    125 pi 273

    pi : pi .

    3.2.25

  • Riemann

    .

    JJ II

    J I

    126 pi 273

    pi : pi .

    3.2.26

  • Riemann

    .

    JJ II

    J I

    127 pi 273

    pi : Sn =n

    k=122k +

    nk=1

    (1)k2k .

    3.2.27

  • Riemann

    .

    JJ II

    J I

    128 pi 273

    pi : pi .

    3.2.28

  • Riemann

    .

    JJ II

    J I

    129 pi 273

    pi : pi .

    4.2.1

  • Riemann

    .

    JJ II

    J I

    130 pi 273

    pi : pi .

    4.2.2

  • Riemann

    .

    JJ II

    J I

    131 pi 273

    pi : pi .

    4.2.3

  • Riemann

    .

    JJ II

    J I

    132 pi 273

    pi : pi .

    4.2.4

  • Riemann

    .

    JJ II

    J I

    133 pi 273

    pi : pi .

    4.2.5

  • Riemann

    .

    JJ II

    J I

    134 pi 273

    pi : | sin(x)|xa 1xa | cos(x)|xa 1xa x 1.

    4.2.6

  • Riemann

    .

    JJ II

    J I

    135 pi 273

    pi : pi t3

    dxx2+x2 pi pi t

    4.2.7

  • Riemann

    .

    JJ II

    J I

    136 pi 273

    pi : pi .

    4.2.8

  • Riemann

    .

    JJ II

    J I

    137 pi 273

    pi : > 0 x1, x2 > | x2x1

    sin(x)x dx | <

    4.2.9

  • Riemann

    .

    JJ II

    J I

    138 pi 273

    pi : pi .

    4.2.10

  • Riemann

    .

    JJ II

    J I

    139 pi 273

    pi : pi .

    4.2.11

  • Riemann

    .

    JJ II

    J I

    140 pi 273

    pi : pi .

    4.2.12

  • Riemann

    .

    JJ II

    J I

    141 pi 273

    pi : pi Cauchy.

    4.2.13

  • Riemann

    .

    JJ II

    J I

    142 pi 273

    pi : pi .

    4.2.14

  • Riemann

    .

    JJ II

    J I

    143 pi 273

    pi : k < 1 limxa(x a)kf (x) = A < +

    baf (x)dx .

    4.2.15

  • Riemann

    .

    JJ II

    J I

    144 pi 273

    pi : pi x = 1t pi pi pipi.

    4.2.16

  • Riemann

    .

    JJ II

    J I

    145 pi 273

    pi : pi .

    4.2.17

  • Riemann

    .

    JJ II

    J I

    146 pi 273

    pi : pi .

    4.2.18

  • Riemann

    .

    JJ II

    J I

    147 pi 273

    pi : pi .

    4.2.19

  • Riemann

    .

    JJ II

    J I

    148 pi 273

    pi : pi .

    4.2.20

  • Riemann

    .

    JJ II

    J I

    149 pi 273

    pi : pi .

    4.2.21

  • Riemann

    .

    JJ II

    J I

    150 pi 273

    pi : .

    4.2.22

  • Riemann

    .

    JJ II

    J I

    151 pi 273

    pi : .

    4.2.23

  • Riemann

    .

    JJ II

    J I

    152 pi 273

    pi : .

    4.2.24

  • Riemann

    .

    JJ II

    J I

    153 pi 273

    pi : .

    4.2.25

  • Riemann

    .

    JJ II

    J I

    154 pi 273

    pi : .

    4.2.26

  • Riemann

    .

    JJ II

    J I

    155 pi 273

    pi : .

    4.2.27

  • Riemann

    .

    JJ II

    J I

    156 pi 273

    pi : .

    4.2.28

  • Riemann

    .

    JJ II

    J I

    157 pi 273

    pi : f (x) = n=0 an pi f (x) pi 2f (x) + f (x) = 0 .

    4.2.29

  • Riemann

    .

    JJ II

    J I

    158 pi 273

    pi : pi |x2| < R.

    4.2.30

  • Riemann

    .

    JJ II

    J I

    159 pi 273

    pi : pi

    4.2.31

  • Riemann

    .

    JJ II

    J I

    160 pi 273

    pi : (x + 1)(x +

    x + 1) = 2x + 2

    x + x

    32 + 1.

    (x + 1)(x x + 1)dx =

    (2x + 2

    x + x

    32 + 1)dx

    =

    2xdx +

    2xdx +

    x

    32dx +

    dx

    = x2 +1x+25x

    52 + x + C.

    1.2.1

  • Riemann

    .

    JJ II

    J I

    161 pi 273

    pi : x2ex

    3dx =

    13

    (x)e(x)dx,

    pi y = (x) = x3, x2ex

    3dx =

    eydy =

    ey

    3 + C =ex

    3

    3 + C

    1.2.2

  • Riemann

    .

    JJ II

    J I

    162 pi 273

    pi : 1

    1 + ex = 1 ex

    1 + ex = 1 (1 + ex )

    1 + ex ,

    1ex + 1dx =

    dx

    (1 + ex )

    1 + ex dx = x ln(1 + ex ) + C

    1.2.3

  • Riemann

    .

    JJ II

    J I

    163 pi 273

    pi :

    x + 22x 1 =

    122x + 42x 1 =

    122x 1 + 52x 1 =

    12 +

    52

    12x 1 .

    x + 22x 1dx =

    (12 +

    52

    12x 1 )dx

    =12

    dx +

    52

    12x 1dx

    =x

    2 +54 ln |2x 1| + C.

    1.2.4

  • Riemann

    .

    JJ II

    J I

    164 pi 273

    pi :

    1sin(x)

    dx =

    12 sin( x2 ) cos(

    x2 )dx

    =

    cos( x2 )sin( x2 ) cos

    2( x2 )d(

    x

    2 ) = 1

    tan( x2 ) cos2( x2 )

    d(x

    2 )

    =

    (tan( x2 ))tan( x2 )

    d(x

    2 ) = ln(tan(x

    2 )) + C,

    kpi < x < kpi + pi, k Z .

    1.2.5pi : x = g(y) = sin(y), x [1, 1] pi y = arcsin(x), (y [ pi2 , pi2 ]) cos(y) 0 .

    1 x2dx =

    1 g(y)2g(y)dy =

    cos(y)(sin(y))dx

    =

    cos2(y)dy =

    1 + cos(2y)2 dy

    =y

    2 +sin(2y)

    4 + c =y

    2 +sin(y) cos(y)

    2 + c

    =arcsin(x)

    2 +x1 x22 + c. (4.1)

  • Riemann

    .

    JJ II

    J I

    165 pi 273

    1.2.6pi : pipi (x a)(b x) < 0 pi a < x < b 0 < xaba < 1.

    xaba = sin

    2(u)

    x = g(u) = a + (b a) sin2(u), u (0, pi2 ),

    b x = (b a) (x a) = (b a) (b a) sin2(u) = (b a) cos2(u), (x a)(b x) = (b a) sin(u) cos(u).

    pi 1(x a)(b x)dx =

    1(g(u) a)(b g(u))du =

    =

    1(b a) sin(u) cos(u) (b a)2 sin(u) cos(u))du

    = 2

    du = 2u + c = 2 arcsin(x ab x

    ) 12+ c.

    1.2.7pi : x = a + (b a) sin2(u)

    (x a)(b x)dx =

    (b a) sin(u) cos(u)(b a)2 sin(u) cos(u)du

    = 2(b a)2

    sin2(u) cos2(u)du =(b a)2

    2

    sin2(2u)du.

  • Riemann

    .

    JJ II

    J I

    166 pi 273

    sin2(2u)du = 12

    sin2(v)dv (2u = v)

    =14

    (1 cos(2v))dv = 14v

    18 sin(2v) + c

    =12u

    18 sin(4u) + c =

    12u

    14 sin(2u) cos(2u) + c.

    =12u

    12 sin(u) cos(u)(1 2 sin

    2(u)) + c.

    pi (x a)(b x) = (b a) sin(u) cos(u)

    I =

    (x a)(b x)dx

    =(b a)2

    2

    sin2(2u)du

    =(b a)2

    4 u (b a)2

    4 sin(u) cos(u)(1 2 sin2(u)) + c =

    =(b a)2

    4 arcsinx a

    b a (b a)24

    (x a)(b x)

    b a (a 2x ab a ) + c

    =(b a)2

    4 arcsinx a

    b a 14 (x a)(b x)(b a 2(x a)) + c

    =(b a)2

    4 arcsinx a

    b a + 12 (x a)(b x)

    (x b + a2

    )+ c.

  • Riemann

    .

    JJ II

    J I

    167 pi 273

    1.2.8pi : ex = y. exdx = dy

    ex

    e2x + 1dx =

    y

    y2 + 11ydy

    =

    1y2 + 1dy.

    y = tan(u) dy = 1cos2(u)du. 1y2 + 1dx =

    1tan2(u) + 1

    1cos2(u)

    du

    =

    du = u + C = arctan(ex ) + C.

    1.2.9pi : x =

    3 tan(u). dx =

    3 1cos2(u)du 1

    3 + x2dx = 1

    31

    tan2(u) + 11

    cos2(u)du

    =13

    du =

    13u + C =

    13

    arctan(x3) + C.

  • Riemann

    .

    JJ II

    J I

    168 pi 273

    1.2.10pi : x =

    5 cosh(u).

    x2 5dx =

    5 sinh(u)5 sinh(u)du

    =

    5 sinh2(u)du = 5

    cosh(2z) 1

    2 du =54 sinh(2u)

    52u

    =125 sinh(u)

    5 sinh(u) 52u

    =xx2 52

    52 ln

    x + x2 55

    + C.

    1.2.11pi : x = 12 (1 cos(u)).

    xx x2

    dx =

    x

    2 12x x2dx

    =

    12 (1 cos(u))

    12 sin(u)

    12 sin(u)du =

    12

    (1 cos(u))du = 12 (u sin(u))

    =12 arccos(1 2x)

    x x2 + C.

  • Riemann

    .

    JJ II

    J I

    169 pi 273

    1.2.12pi : t = ax + b.

    cos(ax + b)dx =1a

    cos(t)dt

    =1a

    sin(t) = 1a

    cos(at + b) + C..

    1.2.13pi : a b

    5x + 3x2 + 2x 3 =

    a

    x 1 +b

    x + 3 ,

    x , 1 x , 3. 5x +3 = a(x +3)+b(x 1). x, x , 1 x , 3 pipi a = 2, b = 3.pi

    5x + 3x2 + 2x 3dx = 2

    1x 1dx + 3

    1x + 3dx

    = 2 ln |x 1| + 3 ln |x + 3| + c = ln |(x 1)2(x + 3)3| + c.

  • Riemann

    .

    JJ II

    J I

    170 pi 273

    1.2.14pi :

    x2 = arctan(t) dx =2

    1+t2dt. 11 + cos(x)dx =

    11 + 1t21+t2

    21 + t2dt

    =

    11+t2+1t2

    1+t2

    21 + t2dt =

    1 + t22

    21 + t2dt

    =

    dt = t + C = tan(

    x

    2 ) + C.

    1.2.15pi : x = 2 tan(u)

    x

    4 + x2dx =

    2 tan(u)2

    cos(u)

    2cos2(u)

    du =

    tan(u)cos(u)

    du

    =

    sin(u)cos2(u)

    du

    =

    (cos(u))

    cos2(u)= 2 1

    cos(u)=4 + x2 + C.

    1.2.16

  • Riemann

    .

    JJ II

    J I

    171 pi 273

    pi : x = t

    1x

    cos(x)dx =

    1tcos(t)2tdt = 2

    cos(t)dt

    = 2 sin(t) + C = 2 sin(x) + C

    1.2.17pi : a, b c

    2x2 + 5x 1x3 + x2 2x =

    a

    x+

    b

    x 1 +c

    x + 2 ,

    x , 0, x , 1 x , 2. pipi a = 12 , b = 2,c = 12 .pi

    2x2 + 5x 1x3 + x2 2x dx =

    12

    1xdx + 2

    1x 1dx

    12

    1x + 2dx

    =12 ln |x | + 2 ln |x 1|

    12 ln |x + 2| + C = ln |(x 1)

    2

    x

    x + 2 | + C.

    1.2.18

  • Riemann

    .

    JJ II

    J I

    172 pi 273

    pi : a, b c x2 + 2x + 3

    (x 1)(x + 1)2 =a

    x 1 +b

    x + 1 +c

    (x + 1)2 ,

    x , 1,1, pipi a = 32 , b = 12 , c = 1.pi

    x2 + 2x + 3

    (x 1)(x + 1)2dx =32

    1x 1dx +

    12

    1x + 1dx

    1(x + 1)2dx

    =32 ln |x 1| +

    12 ln |x + 1| +

    1x + 1 + C = ln |

    (x 1)3x + 1 | + C.

    1.2.19pi : a, b c

    3x2 + 2x 2x3 1 =

    3x2 + 2x 2(x 1)(x2 + x + 1) =

    a

    x 1 +bx + c

    x2 + x + 1 .

    pipi a = 1, b = 2, c = 3.pi

    3x2 + 2x 2x3 1 dx =

    1x 1dx +

    2x + 3x2 + x + 1dx

    = ln |x 1| + 2x + 1

    x2 + x + 1dx + 2 1

    x2 + x + 1dx

    = ln |x 1| + ln(x2 + x + 1) + 2 1

    x2 + x + 1dx

  • Riemann

    .

    JJ II

    J I

    173 pi 273

    1x2+x+1dx pi x

    2 + x + 1 = (x + 12 )2 + 34 =

    34 (y

    2 + 1)pi y = f (x) = 2

    (3) (x +12 ) f

    (x) = 23 . 1x2 + x + 1dx =

    1(x + 12 )

    2 + 34dx =

    f (x)

    34 (f (x)

    2 + 1)dx

    =23

    1y2 + 1dx =

    23

    arctan(y) + c.

    3x2 + 2x 2x3 1 dx = ln |x 1| + ln(x

    2 + x + 1) + 43

    arctan(2x + 1

    3) + C

    1.2.20pi : pi

    P(x)Q(x)dx =

    (x)Q1(x)

    +

    1(x)Q2(x)

    dx. Q1(x)={(x3 1)2,6x2(x3 1)} =

    x3 1. pi (x) = Ax2 + Bx + , Q2(x) = x3 1, 1(x) = x2 + Ex + Z . 1(x3 1)2dx =

    Ax2 + Bx +

    x3 1 +

    x2 + Ex + Z

    x3 1 dx.

    1(x3 1)2 =

    (2Ax + B)(x3 1) 3x2(Ax2 + Bx + )(x3 1)2 +

    x2 + Ex + Z

    x3 1 ,

    (2Ax + B)(x3 1) 3x2(Ax2 + Bx + ) + (x2 + Ex + Z )(x3 1) = 1. A = = = E = 0, B = 13 , Z = 23 . pi

    1(x3 1)2 =

    x

    3(x3 1) 23

    1x3 1dx.

  • Riemann

    .

    JJ II

    J I

    174 pi 273

    1x31dx

    1x31 =

    lx1 +

    mx+nx2+x+1 , pi l =

    13 , m = 13 , n = 23 .

    1x3 1dx =

    13

    1x 1dx

    13

    x + 2

    x2 + x + 1dx

    =13 ln(x 1)

    16 ln(x

    2 + x + 1) 13

    arctan(2x + 1

    3) + C.

    1(x3 1)2dx =

    x

    3(x3 1) +19 ln

    (x2 + x + 1(x 12)

    )+

    233

    arctan(2x + 1

    3) + C

    1.2.21pi :

    x2ex3dx =

    x2(ex )dx = x2ex

    (x2)exdx = x2ex 2

    xexdx

    = x2ex 2

    x(ex )dx = x2ex 2xex + 2

    (x)exdx

    = x2ex 2xex + 2

    1exdx = x2ex 2xex + 2ex + C.

    1.2.22

  • Riemann

    .

    JJ II

    J I

    175 pi 273

    pi :

    aC(x) =

    aeax cos(bx)dx =

    (eax ) cos(bx)dx

    = eax cos(bx)

    eax (cos(bx))dx = eax

    eax (b sin(bx))dx= eax cos(bx) + bS(x) + C1

    pi

    aC(x) bS(x) = eax cos(bx) + C1 (4.2) bC(x) + aS(x) = eax sin(bx) + C2. (4.3)

    (4.2) (4.3)

    C(x) =a cos(bx) + b sin(bx)

    a2 + b2eax + C3 S(x) =

    a sin(bx) b cos(bx)a2 + b2

    eax + C4.

    1.2.23pi : pi pi .

    x2 cos(x)dx =

    x2(sin(x))dx

    = x2 sin(x) 2

    x sin(x)dx = x2 sin(x) 2

    x(cos(x))dx

    = x2 sin(x) + 2x cos(x) 2

    (x) cos(x)dx

    = x2 sin(x) + 2x cos(x) 2 sin(x) + C.

  • Riemann

    .

    JJ II

    J I

    176 pi 273

    1.2.24pi : 1

    x ln(x)dx =

    (ln(x))

    ln(x)dx

    = ln (ln(x)) + C.

    1.2.25pi :

    I =

    ln(x)x

    dx =

    1xln(x)dx

    =

    (ln(x))ln(x)dx = ln2(x)

    ln(x)x

    dx = ln2(x) I.

    2I = ln2(x) + C I = ln(x)x dx = ln2(x)2 + c 1.2.26

    pi : ln(x)dx =

    (x)ln(x)dx

    = x ln(x)

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

    dx = x ln(x) x + C.

  • Riemann

    .

    JJ II

    J I

    177 pi 273

    1.2.27

    pi :

    (x2 1) cos(3x)dx = 13 (x

    2 1) sin(3x)

    sin(3x)2xdx

    =13 (x

    2 1) sin(3x) + 29

    x(cos(3x))dx

    =13 (x

    2 1) sin(3x) + 29x cos(3x) 29

    cos(3x)dx

    =13 (x

    2 1) sin(3x) + 29x cos(3x) 227 sin(3x) + C.

    1.2.28

  • Riemann

    .

    JJ II

    J I

    178 pi 273

    pi :

    I =

    xex cos(x)dx

    =

    x

    (ex

    2 (sin(x) cos(x)))dx

    =xex

    2 (sin(x) cos(x)) 12

    ex sin(x)dx +

    12

    ex (sin(x))dx

    =xex

    2 (sin(x) cos(x)) 12

    ex sin(x)dx +

    12e

    x (sin(x)) 12

    ex sin(x)dx

    =xex

    2 (sin(x) cos(x)) +12e

    x cos(x) + C.

    1.2.29

    pi : pi sin(2x) cos(x) = 12 (sin(3x) + sin(x)) :

    ex sin(2x) cos(x)dx = 12

    ex sin(3x)dx + 12

    ex sin(x)dx.

  • Riemann

    .

    JJ II

    J I

    179 pi 273

    I1 =

    ex sin(3x)dx = 13

    ex (cos(3x))dx

    =13e

    x cos(3x) + 13

    (ex ) cos(3x)dx = 13e

    x cos(3x) + 19

    ex (sin(3x))dx

    =13e

    x cos(3x) + 19ex sin(3x) 19

    (ex ) sin(3x)dx

    =13e

    x cos(3x) + 19ex sin(3x) I1.

    I1 = 110ex sin(3x) 3cos(3x). I2 =

    ex sin(x) = e

    x

    2 (sin(x) cos(x)).

    ex sin(2x) cos(x)dx = 120e

    x sin(3x) 3cos(3x) + ex

    4 (sin(x) cos(x)) + C.

    1.2.30

    pi : pi

    3x212x

    xdx. :

    3x2 12x

    xdx =

    (32x 12x

    32)dx = x

    32 + x

    12 =

    x2 + 1x

    .

  • Riemann

    .

    JJ II

    J I

    180 pi 273

    3x2 12x

    x

    arctan(x)dx = (

    x2 + 1x

    )arctan(x)dx

    =x2 + 1

    xarctan(x)

    x2 + 1

    xarctan(x)dx

    =x2 + 1

    xarctan(x)

    x2 + 1

    x

    11 + x2dx

    =x2 + 1

    xarctan(x) 2(x) + C.

    1.2.31pi : :

    In+1 =

    1(ax2 + bx + c)n+1

    dx =4a

    4ac b2

    ax2 + bx + c a(x + b2a )(ax2 + bx + c)n+1

    dx

    =4a

    4ac b2 In

    (2ax + b)2

    (ax2 + bx + c)n+1dx

    =4a

    4ac b2 In 1

    n(4ac b2)

    (2ax + b)(

    1(ax2 + bx + c)n

    )dx

    pi

    =4a

    4ac b2 In +1

    n(4ac b2)2ax + b

    (ax2 + bx + c)n 2an(4ac b2)

    1(ax2 + bx + c)n

    dx

    =4a

    4ac b2 In +1

    n(4ac b2) (2ax + b)1

    (ax2 + bx + c)n 2an(4ac b2)n In

  • Riemann

    .

    JJ II

    J I

    181 pi 273

    1.2.32pi : (1 + x

    1 x) 12dx =

    1 + x(1 x)(1 x)dx

    =

    1(1 x2)

    dx +

    x

    (1 x2)dx.

    pi

    1(1x2)dx.

    x(1 x2)

    dx = 12

    (1 x2)(1 x2)

    dx =

    = 12 1

    ydy (y = 1 x2)

    = y + c = 1 x2 + c.

    pi

    1(1x2)dx = arcsin(x). (1 + x

    1 x) 12dx = arcsin(x)

    1 x2 + c

  • Riemann

    .

    JJ II

    J I

    182 pi 273

    1.2.33pi :

    R

    (x,ax2 + bx + c

    )dx a < 0 c > 0

    4 3x x2 = t(x + 4) : 1

    4 3x x2dx =

    10t(1+t2)25t1+t2

    dt =

    = 2 1

    1 + t2dt = 2 arctan(t) + C

    = 2 arctan1 xx + 4

    + C.

    1.2.34pi :

    R

    (x,ax2 + bx + c

    )dx a > 0

    x2 x 1 = t x : 1

    x +x2 x 1

    dx =

    2t2 2t + 2t(2t 1)2 dt =

    =

    (2t 32t 1 +

    3(2t 1)2

    )dt

    = 2 ln(|t |) 32 ln(|2t 1|) 32

    12t 1

    = 2 ln(|x +x2 x + 1|) 32 ln(|2x 1 + 2

    x2 x + 1|)

    32 ln(1

    2x 1 + 2x2 x + 1) + C.

  • Riemann

    .

    JJ II

    J I

    183 pi 273

    1.2.35

  • Riemann

    .

    JJ II

    J I

    184 pi 273

    pi : pi pi pi . pi Pn pi.. U (f, Pn) .

    1n

    nk=1

    nek =

    1n

    ne +

    1n

    ne2 + + 1

    nnen

    =1ne

    1n +

    1ne

    2n + + 1

    ne

    nn

    [0,1], Pn = {x0 = 0, x1 = 1n , x1 = 1n , , xn = 1} f : R f (x) = ex . 10 exdx = [ex ]10 = e e0 = e 1.pi U (f, Pn). U (f, Pn) = 1n

    nk=1

    nek limn U (f, Pn) = limn 1n

    nk=1

    nek

    e 1 = limn 1nn

    k=1nek.

    2.2.1

  • Riemann

    .

    JJ II

    J I

    185 pi 273

    pi : Pn limn[U (f, Pn) L(g, Pn)] = 0. f R([a, b]) pi Pn

    bafdx = limn L(f, Pn).

    pi P n bahdx = limn U (h, Pn). Qn = Pn P n

    ( Qn pi Pn ) pi

    L(f, Pn) L(f, Qn) L(g, Qn) U (g, Qn) U (h,Qn) U (h, P n),pi f (t) g(t) h(t). 0 U (g, Qn)L(f, Qn) U (h, P n)L(f, Pn) limn[U (h,Qn)L(f, Qn)] =

    bahdx b

    afdx = 0. pi

    limn[U (g, Qn) L(g, Qn)] = 0

    g R([a, b]) bagdx limn U (g, Qn) =

    bafdx.

    2.2.2

  • Riemann

    .

    JJ II

    J I

    186 pi 273

    pi : f , 0 ba|f (t)|dt pi .

    f , 0 pi t0 [a, b] : |f (t0)| = , 0. pi f pi pi U t0 |f (t)| > 2 , t U . f , 0 ba|f (t)|dt

    U|f (t)|dt 2 U > 0.

    2.2.3

  • Riemann

    .

    JJ II

    J I

    187 pi 273

    pi : Pn f (x) = 1x U ( 1x , Pn) = 1 +

    12 + 1n1 L( 1x , Pn) = 12 + 1n .

    Pn = {1 < 2 < ... < n} [1, n].

    L(f, Pn) =n

    k=2mk(tk tk1) =

    nk=2

    mk =n

    k=2

    1k,

    U (f, Pn) =n

    k=2Mk(tk tk1) =

    nk=2

    Mk =n

    k=2

    1k 1 .

    2.2.4

  • Riemann

    .

    JJ II

    J I

    188 pi 273

    pi : sin x [a, b] -. Pn = {a, a+h, a+2h, a+nh = b}, h = ban . Riemann, L(sin x, Pn). limn L(sin x, Pn) =

    ba

    sin xdx,

    L(sin x, Pn) =n

    k=1sin(a + (k 1)h)h = h

    nk=1

    sin(a + (k 1)h) = h cos(a h2 ) cos(b h2 )2 sin( h2 )

    .

    limh0 h2 sin( h2 ) = 1, limn L(sin x, Pn) = ba

    sin xdx = cos b cosa.

    2.2.5

  • Riemann

    .

    JJ II

    J I

    189 pi 273

    pi : [a, b] -. pipi ) m , 1, ) m = 1.) Pn = {a, ah, ah2, ahn}, h = n

    ba . h 1

    n . Riemann Pn.

    Sn(f, Pn) = (ah a)am + (ah2 ah)amhm + + (ahn ahn1)amh(n1)m= am+1(h 1) + am+1hm+1(h 1) + + am+1h(n1)(m1)(h1)

    = am+1(h 1){1 + hm+1 + (hm+1)n1}= am+1(h 1) (h

    m+1)n 1hm+1 1 {1 + h

    m+1 + (hm+1)n1} = h 1hm+1 1 (b

    m+1 am+1).

    limn1 h1hm+11 =1

    m+1 . baxmdx = limn Sn(f, Pn) = b

    m+1am+1m+1 .

    ) m = 1 m + 1 = 0 Sn(f, Pn) = n(h 1). pi ahn = b n = ln blnalnh , bax1dx = lim

    nSn(f, Pn) = limh1h 1lnh

    (ln b lna).

    2.2.6

  • Riemann

    .

    JJ II

    J I

    190 pi 273

    pi :

    Pn = {a, a + b an

    , a +2(b a)

    n, , a + n(b a)

    n= b}.

    ||Pn || = ban 0. Riemann Pn

    S(Pn , f,) =n

    k=1f(a + k

    b an

    )||Pn ||.

    limnS(Pn , f,) = limn

    nk=1

    f(a + k

    b an

    )||Pn || =

    baf (x)dx.

    2.2.7

  • Riemann

    .

    JJ II

    J I

    191 pi 273

    pi : limn bann

    k=1 f(a + k ban

    )=

    baf (x)dx. a = 0,

    b = 1

    limn

    1n

    ni=1

    f (i

    n).

    2.2.8

  • Riemann

    .

    JJ II

    J I

    192 pi 273

    pi : an

    an =1n{ 11 + 1n

    + + 11 + nn}.

    pipi pi f (x) = 1x+1 x =1n , , nn . n ,1

    n 0 nn 1. [0,1] pi f pi . Pn = {0, 1n , 2n , , nn = 1} pi limn ban

    nk=1 f

    (a + k ban

    )=

    baf (x)dx,

    limnan = limn{

    1n{ 11 + 1n

    + + 11 + nn}} =

    10

    11 + x dx

    2.2.9

  • Riemann

    .

    JJ II

    J I

    193 pi 273

    pi : f (x) = ex R [0,1].

    Pn = {x0 = 0, x1 = h, x2 = 2h, , xn = nh = 1},pi h = 1n . = {0, h,2h, , (n 1)h} 1

    0exdx = lim

    n

    h n1k=0

    f (kh)

    = limnhe0 n1

    k=0eh

    k

    = lim

    n

    (heh

    n 1eh 1

    )= (e 1) lim

    nh

    eh 1 = (e 1).

    2.2.10

  • Riemann

    .

    JJ II

    J I

    194 pi 273

    pi : C 0 < C < f (c). pi f pi > 0 x [c , c + ] [a, b] f (c) > C. pi b

    af (x)dx > 0 =

    ca

    f (x)dx + c+c

    f (x)dx + bc+

    f (x)dx 0 + 2C + 0 > 0.

    c = a ( c = b) pi [a, a + ] ( [b , b]).

    2.2.11

  • Riemann

    .

    JJ II

    J I

    195 pi 273

    pi :) 1

    0x2dx =

    x3

    3 |10 =

    13

    03 =

    13

    ) 10x3dx =

    x4

    4 |10 =

    14

    04 =

    14

    ) pi0

    cos(x)dx = sin(x)|pi0 = sin(pi) sin(0) = 1 0 = 1.

    ) 11

    (2x2 x3)dx = 2 11x2dx

    11x3dx = 2x

    3

    3 |11

    x4

    4 |11 = 2(

    13

    13 ) (

    14

    14 ) =

    43

    ) 32ex/2dx = 2ex/2|32 = 2(e3/2 e1) =

    2e(1 e1/2).

    2.2.12

  • Riemann

    .

    JJ II

    J I

    196 pi 273

    pi : f (x) = x + pi g(x) = sin(x) x [0, pi]. f g pi g(x) 0 [0, pi]. pi pi pi pi [0, pi] pi

    0(x + pi) sin(x)dx = ( + pi)

    pi0

    sin(x)dx

    2.2.13

  • Riemann

    .

    JJ II

    J I

    197 pi 273

    pi : f (x) = 1x g(x) =sin(x) x [a, b]. pi [a, b] b

    a

    sin(x)x

    dx =1a

    a

    sin(x)dx +1

    b

    sin(x)dx.

    pi a

    sin(x)dx = cos(a) cos( ), b

    sin(x)dx = cos( ) cos(b) | a

    sin(x)dx | 2,| b

    sin(x)dx | 2,

    | ba

    sin(x)x

    dx | 2b+2a 2a+2a=4a.

    2.2.14

  • Riemann

    .

    JJ II

    J I

    198 pi 273

    pi : , f (x) = g(x) x3 x2 x + 1 = x + 1. (1,0), (0,1), (2,3). g(x) f (x) 1 x 0 f (x) g(x) 0 x 2.

    E =

    21|f (x)g(x)|dx =

    01

    [(x3x2x+1)(x+1)]dx+ 20

    [(x+2)(x3x2x+1)]dx = 3712

    2.2.15

  • Riemann

    .

    JJ II

    J I

    199 pi 273

    pi : pi 2pi0

    1 sin(x)dx.

    E =

    2pi0

    (1 sin(x))dx =

    2pi0

    1 + cos(x + pi2 )dx,

    t = x + pi2

    E =

    5pi4

    pi2

    1 + cos(2t)dt = 2

    2 5pi

    4

    pi2

    | cos(t)|dt = 22[ pi

    2

    pi4

    cos(t)dt 5pi

    4

    pi2

    cos(t)dt] = 4(2).

    2.2.16

  • Riemann

    .

    JJ II

    J I

    200 pi 273

    pi : pi y = x2, x + y = 2, x2 = 2 x x = 1 x = 4. (1,1) (4,2). 0 x 1 pi pi 1 x 4 pi pi.

    E =

    10

    [x (x)]dx +

    41

    [(2 x) (x)]dx = 92 . pi pi y

    E =

    12

    [(2 y) y2]dy = 92 .

    2.2.17

  • Riemann

    .

    JJ II

    J I

    201 pi 273

    pi : 0 pi pi pi (3, 2pi3 ). (3, 2pi3 ) , pi (0,0) (0, pi2 ) .

    E = 2 2pi3

    0(2 2 cos())2d 12

    2pi3

    pi2

    (6 cos())2d

    =

    2pi3

    0(6 8 cos() + 2 cos(2))d

    2pi3

    pi2

    (1 + cos(2))d

    = [6 8 sin() + sin(2)]2pi/30 18[ +sin(2)

    2 ]2pi/3pi/2 = pi.

    2.2.18

  • Riemann

    .

    JJ II

    J I

    202 pi 273

    pi : pi y = x2 pi x = t2,y = 2t, t [0,1]

    S =

    10

    (2t)2 + 4dt = 2

    t2 + 1dt

    = 2[12 tt2 + 1 + 12 ln |t +

    t2 + 1|]10 =

    2 + ln(1 +

    2).

    2.2.19

  • Riemann

    .

    JJ II

    J I

    203 pi 273

    pi :

    S =

    2pi0

    x (t)2 + y(t)2dt =

    2pi0

    ( sin(t) sin(2t))2 + (cos(t) + cos(2t))2dt

    =

    2pi0

    2(1 + cos(t))dt = 2

    2pi0

    cos2(

    t

    2 )dt

    = 2 2pi0

    | cos( t2 )|dt = 2 pi0

    cos(t

    2 )dt 2 2pipi

    cos(t

    2 )dt = 8

    2.2.20

  • Riemann

    .

    JJ II

    J I

    204 pi 273

    pi : pi limn bann

    k=1 f(a + k ban

    )=

    baf (x)dx. -

    41

    (2x3 5x)dx = limn

    3n

    ni=1

    f(1 + 3i

    n

    )= lim

    n3n

    ni=1

    [2(1 + 3i

    n

    )3 5

    (1 + 3i

    n

    )]

    = limn

    3n

    ni=1

    [3 + 3 i

    n+ 54 i

    2

    n2+ 54 i

    3

    n3

    ]

    = limn

    3nni=1

    (3) + 9n2

    ni=1

    i +162n3

    ni=1

    i2 +162n4

    ni=1

    i3

    = limn

    3n (3)n + 9n2 n(n + 1)2 + 162n3 n(n + 1)(2n + 1)6 + 162n4(n(n + 1)

    2

    )2= lim

    n

    {9 + 921(1 +

    1n) + 27(1 + 1

    n)(2 + 1

    n) +

    812 (1 +

    1n)2

    }= 9 + 92 + 27 2 +

    812 = 90

    2.2.21

  • Riemann

    .

    JJ II

    J I

    205 pi 273

    pi : u = 3x + 4 du = 3dx. pi x = 0, u = 4 x = 4,u = 16. 4

    0

    3x + 4dx = 13

    146udu =

    1323 [u

    3/2]146

    =29 (16

    3/2 43/2) = 1129 .

    2.2.22

  • Riemann

    .

    JJ II

    J I

    206 pi 273

    pi : pi y = sin(x) y = cos(x) [0, pi/2] pi/4,

    (2)/2. cos(x) sin(x) 0 x pi/4 cos(x) sin(x)

    pi/4 x pi/2. pi

    E =

    pi/20

    | cos(x) sin(x)|dx

    =

    pi/40

    (cos(x) sin(x))dx + pi/2pi/4

    (sin(x) cos(x))dx

    = [sin(x) + cos(x)]pi/40 + [ cos(x) sin(x)]pi/2pi/4= (

    12+

    12 0 1) + (0 1 + 1

    2+

    12) = 2

    2 2.

    2.2.23

  • Riemann

    .

    JJ II

    J I

    207 pi 273

    pi : pi pi pi pi E = 2pi

    ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t), l t m.

    e = a2b2a2 < 1 e cos(t) = u

    E = 2piab pi0

    sin(1 e2 cos2(t))dt = 2pi ab

    e

    ee

    1 u2du

    = 2piab[ arcsin(e)e

    +1 e2].

    2.2.24

  • Riemann

    .

    JJ II

    J I

    208 pi 273

    pi : pi pi pi pi E = 2pi

    ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t), l t m.

    E = 2pi 2pi0

    a2(1 cos(t))2(1 cos(t))dt= 4pia2

    2pi0

    (1 cos(t)) sin( t2 )dt =643 pia

    2.

    2.2.25

  • Riemann

    .

    JJ II

    J I

    209 pi 273

    pi : pi pi pi pi E = 2pi

    ml|f (t)|f (t)2 + g(t)2dt pi x = f (t), y = g(t), l t m.

    pi pipi l, m pipi pi x . y = 0 pi t = 0 t = 3. x = 0 x = 3. pi x- (0,0) (3,0). pi pi pi x - .

    E = 2pi 30

    |y(t)|x (t) + y(t)dt= 2pi

    30

    t3 (t2 3)(4t2 + (t2 1)2)dt

    = 2pi 30

    t3 (t2 3)(1 + t2)dt = 3pi.

    2.2.26

  • Riemann

    .

    JJ II

    J I

    210 pi 273

    pi : pi pi pi f , g, f (x) g(x) 0, x [a, b] V = pi b

    a{f (x2) g(x)2}dx. pipi

    pi

    V = pi

    21{(x + 2)2 x4}dx = 72pi5

    2.2.27

  • Riemann

    .

    JJ II

    J I

    211 pi 273

    pi : pi pi pi - f , x [a, b] V = pi b

    af (x)2dx.

    V = pi

    aa

    y2dx = pi

    aa

    b2(1 x

    2

    a2

    )dx =

    43piab

    2

    2.2.28

  • Riemann

    .

    JJ II

    J I

    212 pi 273

    pi : pi pi pi - x = g(t), y = f (t), t [t1, t2] V = pi

    t2t1f (t)2g(t)dt.

    V = 2pi a0

    y2dx

    x(t1) = 0 t = pi2 x(t2) = a t = 0

    V = 2pi 2pi0

    y2dx = 2pi 0pi/2

    a2 sin6(t)(3a cos2(t) sin(t))

    = 6pia3[ pi/20

    sin7(t)dt pi/20

    sin9(t)dt]

    = 6pia3(674523 +

    89674523

    )=

    32105pia

    3.

    2.2.29

  • Riemann

    .

    JJ II

    J I

    213 pi 273

    pi : [0,1] 10 . xi = i10 , i = 1, ,10 : f (0) = 1, f (x1) = 1.00005, f (x2) = 1.00080, f (x3) = 1.00404, f (x4) =1.01272, f (x5) = 1.03078, f (x6) = 1.026283, f (x7) = 1.11360, f (x8) = 1.18727, f (x9) =1.28690, f (x10) = 1.41421.

    T10 =120 [f (0) + 2f (x1) + + 2f (9) + f (1)] = 1.09061

    .

    2.2.30

  • Riemann

    .

    JJ II

    J I

    214 pi 273

    pi : [0,1] 10 . xi = i10 , i = 1, ,10 : f (0) = 1, f (x1) = 1.00005, f (x2) = 1.00080, f (x3) = 1.00404, f (x4) =1.01272, f (x5) = 1.03078, f (x6) = 1.026283, f (x7) = 1.11360, f (x8) = 1.18727, f (x9) =1.28690, f (x10) = 1.41421.

    S =130 [f (0) + {f (

    110 ) + f (

    310 ) + f (

    510 ) + f (

    710 ) + f (

    910 )}

    +{f ( 210 ) + f (410 ) + f (

    610 ) + f (

    810 )} + f (1)]

    =32.68473

    30 = 1.08949.

    2.2.31

  • Riemann

    .

    JJ II

    J I

    215 pi 273

    pi : nn+3 < 1 n

    2n (n+3) 1 3 > 1

    n=1

    n+2n3 .

    3.2.5

  • Riemann

    .

    JJ II

    J I

    220 pi 273

    pi : . an = 3n

    n10 , an , 0 N.limn | an+1an | = limn |

    3n+1(n+1)10

    3nn10

    | = limn 3n10(n+1)10 = 3 limn n10

    (n+1)10 = 3 limn1

    ( n+1n )10 =

    3 limn 1(1+ 1n )10 = 3 > 1. n=1 3nn10 pi.

    3.2.6

  • Riemann

    .

    JJ II

    J I

    221 pi 273

    pi : . an = n5n/2 0 limn |an |1n =

    limn a1nn = limn

    (n

    5n/2) 1n= limn n

    1n

    512= 1

    5limn n1/n = 15 < 1.

    n=1

    n5n/2

    .

    3.2.7

  • Riemann

    .

    JJ II

    J I

    222 pi 273

    pi : . an = 9n

    n! , 0 | an+1an | = |9n+1(n+1)!9nn!| =

    | 9n+1n!9n (n+1)! | = | 9n+1 | = 9n+1 0 n N. pi pi n=1

    9nn! .

    3.2.8

  • Riemann

    .

    JJ II

    J I

    223 pi 273

    pi : . an = nn

    (3n+1)n limn |an |1n =

    limn a1nn = limn

    (nnn

    n(3n+1)n)= limn |n||3n+1| = limn

    n3n+1 = limn

    13n+1n

    = limn 13+ 1n=

    13 . pi pi

    n=1

    nn

    (3n+1)n .

    3.2.9

  • Riemann

    .

    JJ II

    J I

    224 pi 273

    pi : . an = 1+cos2(nx)

    2n |an |1n = a

    1nn =

    n1+cos2(nx)

    2 . 12 n

    an 12 n

    2, pi pi pi pi

    pi n1 + cos2(nx) n

    2. pi

    nan 12 .

    n=11+cos2(nx)

    2n .

    3.2.10

  • Riemann

    .

    JJ II

    J I

    225 pi 273

    pi : n=1 (n!)nnn2 an = (n!)nnn2 . n=1 an . an > 0 n N |an | = an. .limn |an |1/n = limn

    n(n!)nnnnn = limn

    |n!|nn = limn

    n!nn = p R

    {,+}. 0 p < 1 n=1 an an = (n!)nnn2 0 n .pi pipi p = 0 limn n!nn = p = 0. bn = n!nn .

    n=1 bn pi limn bn = 0.

    . bn , 0, n N.bn+1bn

    = |(n+1)!

    (n+1)n+1n!nn

    | = | nn (n+1)n!(n+1)n+1n! | = nn

    (n+1)n =(

    11+ 1n

    )n= 1(1+ 1n )n

    1e < 1 n .

    n=1 an

    limn (n!)n

    nn2= 0.

    3.2.11

  • Riemann

    .

    JJ II

    J I

    226 pi 273

    pi : n=1 12n1 , pi Sn = 1 +

    12 +

    122 +

    123 + +

    12n 1 =

    1 ( 12 )n1 ( 12 )n

    112= 2 n ,

    n pi 12 .

    n=11

    2n1 = 2. pi0 11+2n1 12n1 . pi pi

    n=1

    11+2n1 .

    3.2.12

  • Riemann

    .

    JJ II

    J I

    227 pi 273

    pi : n=1 1n , ( ) pi n=1

    1n = +

    0 1n 2n1n pi

    n=1

    2n1n

    pi.

    3.2.13

  • Riemann

    .

    JJ II

    J I

    228 pi 273

    pi : bn = 12n1 an =1n .

    n=1

    1n

    pi.

    limn

    anbn

    = limn

    1n1

    2n1= lim

    n(2 1n) = 2.

    n=1 12n1 pi.

    3.2.14

  • Riemann

    .

    JJ II

    J I

    229 pi 273

    pi : a 0 + 1na , 0. a = 1 +. a > 0 f (x) = 1xa [1,+) a , 1

    limnF (n) = limn

    n1

    tadt = limn[

    11 a n

    1a 11 a ] ={ a < 1

    11a a > 1

    pi , , a > 1.pi pi pipi n=1 1na 1a1 + 1 = aa1 .

    3.2.15

  • Riemann

    .

    JJ II

    J I

    230 pi 273

    pi : an = n2n .

    n|an | = n

    | n2n | =

    nn

    2 12 n .

    n=1 n2n . 3.2.16

  • Riemann

    .

    JJ II

    J I

    231 pi 273

    pi : an = nn

    n! , 0. .

    |an+1an

    | = |(n+1)n(n+1)!nn

    n!

    | = |n!(n + 1)n+1

    n!(n + 1)nn | = |(n + 1)n

    nn| =

    (n + 1n

    )n=

    (1 + 1

    n

    )n e > 1,

    n . pi n=1 nnn! pi.

    3.2.17

  • Riemann

    .

    JJ II

    J I

    232 pi 273

    pi : an =(1 + 1n

    )n2. .

    |an | 1n = a1nn =

    n

    (1 + 1

    n

    )n2=

    (1 + 1

    n

    )n e > 1,

    n . pi n=1 (1 + 1n )n2 pi.

    3.2.18

  • Riemann

    .

    JJ II

    J I

    233 pi 273

    pi : 3n 2n 13n 12n . 12n 0 12n + 13n 22n , n N.pi n=1 22n .

    n

    | 22n | =

    n

    22n =

    n22

    12 < 1 n .

    n=1 ( 12n + 13n ) . piSn =

    (12 +

    122 +

    123 + +

    12n

    )+

    (13 +

    132 +

    133 + +

    13n

    )=

    12

    ( 12 )n 1

    12 1

    +13

    ( 13 )n 1

    13 1

    32 n .

    3.2.19

  • Riemann

    .

    JJ II

    J I

    234 pi 273

    pi : n=1 n1357(2n1) . an =n

    1357(2n1) , 0. .

    |an+1an

    | = an+1an

    =

    n+11357(2n+1)

    n1357(2n1)

    =n + 1

    n(2n + 1) 0 n .

    n=1 n1357(2n1) limn n1357(2n1) = 0.

    3.2.20

  • Riemann

    .

    JJ II

    J I

    235 pi 273

    pi : 0 an 9 0 an10n 910n . n=1 910n = 9n=1 110n = 9 19 = 1, pi = 110 < 1.pi pi n=1 an10n pi [0,1].

    3.2.21

  • Riemann

    .

    JJ II

    J I

    236 pi 273

    pi : an = 5n

    7nn32, 0 n N .

    |an+1an

    | =5n+1

    7n+1(n+1)32

    5n

    7n (n)32

    =5n+17nn 32

    5n7n+1(n + 1) 32=

    5n 327(n + 1) 32

    =

    57

    ( nn + 1

    ) 32=57

    1n+1n

    32 = 57 11 + 1n

    32 57 < 1 n .

    3.2.22

  • Riemann

    .

    JJ II

    J I

    237 pi 273

    pi : an = sin( 1n ).

    n=11n bn =

    1n . -

    . bn > 0, an n N anbn =sin( 1n )

    1n

    1 > 0.pi n=1 1n = + pi pi

    n=1 sin(1n ) pi.

    3.2.23

  • Riemann

    .

    JJ II

    J I

    238 pi 273

    pi : f (x) = 1x(ln(x))a , a > 0 x [2,+) pi pipi- . a , 1

    limnF (n) = limn

    n2

    1t(ln(t))a

    dt =1

    1 a limn[ln(n)1a ln(2)1a]

    =

    { 1a1 (ln(2))

    1a , a > 1+, a < 1.

    a = 1 limn F (n) = limn[ln(ln(n)) ln(ln(2))] = +.

    3.2.24

  • Riemann

    .

    JJ II

    J I

    239 pi 273

    pi : n=1(1)n n2+12n3+n1 . an = n

    2+12n3+n1 . an an 0 n . pi .

    pi pi , n=1 n2+12n3+n1 . pi

    n2 + 12n3 + n 1

    n2

    2n3 + n 1 2n2

    2n3 + 2n3 + 2n3 =14n .

    n=1 n2+12n3+n1 pi n=1(1)n n2+12n3+n1 -.

    3.2.25

  • Riemann

    .

    JJ II

    J I

    240 pi 273

    pi : n=1 (1)n+12n . an =

    12n . an =

    12n an 0 n . pi .

    pi pi , n=1 12n . r = 1/2 < 1 pi.

    3.2.26

  • Riemann

    .

    JJ II

    J I

    241 pi 273

    pi : Sn =n

    k=12+(1)k

    2k , Sn =nk=1

    22k +

    nk=1

    (1)k2k . limn

    nk=1

    12k1 =

    11 12

    = 2. -

    Tn =n

    k=1(1)k2k

    Tn =n

    k=1(12 )

    k = 12 + (12 )

    2 + + (12 )n

    = 12 [1 + (12 ) + + (

    12 )

    n1] = 12( 12 )n 1 12 1

    .

    Tn =( 12 )1

    3 , limn Tn =13 limn( 12 ) 13 = 13

    pin=1

    2 + (1)n2n =

    n=1

    22n +

    n=1

    (1)n2n = 2

    13 =

    53 .

    3.2.27

  • Riemann

    .

    JJ II

    J I

    242 pi 273

    pi : limn 1n = 0, limn ln(n+1n ) = ln(1) =

    0, 1n > ln(n+1n ) >

    1n+1 . ( c Euler,

    c = 0.577215).

    3.2.28

  • Riemann

    .

    JJ II

    J I

    243 pi 273

    pi : s > a k , 1

    lims

    sa

    dx

    xk=

    11 k lims[s

    1k a1k] ={ 1

    1ka1k k > 1

    + k < 1. k > 1 pi k < 1. k = 1

    lims

    sa

    dx

    x= lim

    s[ln(s) ln(a)] = +.

    4.2.1

  • Riemann

    .

    JJ II

    J I

    244 pi 273

    pi : 0

    11 + x2dx = limt

    0t

    11 + x2dx = limt arctan(t) =

    pi

    2 .

    4.2.2

  • Riemann

    .

    JJ II

    J I

    245 pi 273

    pi :

    11 + x2dx =

    0

    11 + x2dx +

    0

    11 + x2dx

    = limt

    0t

    11 + x2dx + lims

    s0

    11 + x2dx

    = limt arctan(t) + lims arctan(s) = (

    pi

    2 ) +pi

    2 = pi.

    4.2.3

  • Riemann

    .

    JJ II

    J I

    246 pi 273

    pi : )

    limt

    t1

    x1 + x2

    dx = limt[

    1 + t2 2] = +.

    .)

    limt

    t0

    cos(t) = limt sin(t).

    limt sin(t) pi. .)

    limt

    t0exdx = lim

    t[1 et] = 1.

    .)

    limt

    t1

    ln(x)x

    dx = limt(

    12 ln(t)

    2) = +. .

    4.2.4

  • Riemann

    .

    JJ II

    J I

    247 pi 273

    pi : ex2 [0,+]. 1 e

    x2dx t [1,+] et2 < et F (t) = t1 ex2dx t1 e

    xdx 1 e

    xdx

    limt

    t0exdx = lim

    t[1 et] = 1.

    1 e

    x2dx . pi pi 0

    ex2dx =

    10ex

    2dx +

    1

    ex2dx,

    10 e

    x2dx pi 0 e

    x2dx .

    4.2.5

  • Riemann

    .

    JJ II

    J I

    248 pi 273

    pi : | sin(x)|xa 1xa | cos(x)|xa 1xa x [a, b]. -

    1

    1xa dx a > 1

    1

    sin(x)xa dx

    1

    cos(x)xa dx

    pi a > 1.pi t > 1 t

    1

    sin(x)xa

    dx =

    [cos(x)

    xa

    ]t1 a

    t1

    cos(x)x1+a

    dx.

    t1

    cos(x)x1+a dx 1 + a > 1 a > 0 pi

    1

    sin(x)xa

    dx = limt

    [cos(x)

    xa

    ]t1 a lim

    t

    t1

    cos(x)x1+a

    dx = cos(1) a 1

    cos(x)x1+a

    .

    pi a > 0 .

    4.2.6

  • Riemann

    .

    JJ II

    J I

    249 pi 273

    pi : pi t3

    dxx2+x2 , t > 3 t

    3

    dx

    x2 + x 2 = 13

    t3

    dx

    x + 2 +13

    t3

    dx

    x 1 =13 ln(

    52t 1t + 2 ).

    t 3

    dx

    x2 + x 2 =13 ln(

    52 ).

    4.2.7

  • Riemann

    .

    JJ II

    J I

    250 pi 273

    pi : pi 1

    x

    (1 + x2)dx = 1

    xd

    1(1 + x2)

    = limx

    ( x

    1 + x 12

    )+

    1

    dx

    2x(1 + x)

    =12 +

    12

    1

    dx

    2x(1 + x)

    =12 +

    12

    1

    2tdtt(1 + t2)

    =12 + limx(arctan

    (x) pi4 )

    =12 +

    pi

    2 pi

    4 =12 +

    pi

    4 .

    4.2.8

  • Riemann

    .

    JJ II

    J I

    251 pi 273

    pi : Cauchy a < x1 < x2 x2x1

    sin(x)x

    dx = x2x1

    1xd cos(x) =

    cos(x1)x1

    cos(x2)x2

    x2x1

    cos(x)x2

    dx.

    pi

    | x2x1

    sin(x)x

    dx | 1x1

    +1x2

    + | x2x1

    cos(x)x2

    dx |

    1x1

    +1x2

    +

    x2x1

    1x2

    dx =2x1

    < . (4.4)

    pi = 2 pi Cauchy a

    sin(x)x dx

    .

    4.2.9

  • Riemann

    .

    JJ II

    J I

    252 pi 273

    pi : f (x) = 1x4+1

    0 < f (x) = 11 + x4 1) pi

    a

    11+x4dx .

    4.2.10

  • Riemann

    .

    JJ II

    J I

    253 pi 273

    pi : f (x) = 1(1+x3)1/3

    f (x) =1

    (1 + x3)1/3>

    11 + x = g(x),

    x [0,+]. a

    11+x dx = + pi

    a

    1(1+x3)1/3dx = +

    4.2.11

  • Riemann

    .

    JJ II

    J I

    254 pi 273

    pi : f (x) = x22x4x2+1 g(x) =x2

    x4 =1x2 (pi pi

    f pi). pi limx f (x)g(x) =12

    1

    dxx2 < + pi

    0

    x2

    2x4x2+1dx < +

    4.2.12

  • Riemann

    .

    JJ II

    J I

    255 pi 273

    pi : Cauchy > 0 x1, x2 > | x2

    x1x sin(x4)dx | < > 0. 0 x sin(x4)dx = 10 x sin(x4)dx +

    1 x sin(x4)dx x2

    x1

    x sin(x4)dx =14

    x42x41

    sin(t)tdt

    = 14cos(t)

    t|x42x41 18

    x42x41

    cos(t)t3/2

    dt

    = 14[cos(x22 )x22

    cos(x21 )

    x21

    ] 18

    x42x41

    cos(t)t3/2

    dt.

    | x2x1

    x sin(x4)dx | 14 (1x22

    +1x21

    ) +18

    x42x41

    dt

    t3/2

    =14 (

    1x22

    +1x21

    ) 14 (1x22

    1x21

    ) =12x21

    < . (4.5)

    = 12 pi Cauchy pi 0 x sin(x

    4)dx,.

    4.2.13

  • Riemann

    .

    JJ II

    J I

    256 pi 273

    pi : 1(xa)k (a, b]. a < t < b k , 1 b

    t

    dx

    (x a)k =1

    1 k[

    1(b a)k1

    1(t a)k1

    ],

    pilimta

    bt

    dx

    (x a)k ={ 1

    1k (b a)1k k < 1+ k > 1.

    ba

    dx(xa)k k < 1 pi k > 1. k = 1

    limta bt

    dxxa = limta[ln |b a| ln |t a|] = , pi.

    4.2.14

  • Riemann

    .

    JJ II

    J I

    257 pi 273

    pi : pi ca

    dx(xa)(bx) ,

    bc

    dx(xa)(bx) .

    pi

    limxa

    (x a)1/2(x a)(b x) =

    1(b a) ,

    limxb

    (b x)1/2(x a)(b x) =

    1(b a) .

    pi x = a cos2(t) + b sin2(t) ba

    dx(x a)(b x) = 2

    pi/20

    dt = pi.

    4.2.15

  • Riemann

    .

    JJ II

    J I

    258 pi 273

    pi : x = 1t 10

    sin(1x)dx =

    l

    sin(t)t2

    dt.

    l

    sin(t)t2 dt pi

    | l

    sin(t)t2

    dt | l

    1t2dt < +.

    pi 10 sin(

    1x )dx .

    4.2.16

  • Riemann

    .

    JJ II

    J I

    259 pi 273

    pi : pi0

    dx

    sinp(x)=

    pi/20

    dx

    sinp(x)+

    pipi/2

    dx

    sinp(x).

    pi x = pi t pipi/2

    dxsinp(x)

    pi/20

    dxsinp(x) . pi limx0+ x

    p 1sinp(x) = 1 pi pi,pi

    , 0 < p < 1.

    4.2.17

  • Riemann

    .

    JJ II

    J I

    260 pi 273

    pi :

    (x) = 10tx1etdt +

    +1

    tx1etdt.

    pipi ) x 1 )x < 1 ) x 1 : pipi 10 tx1etdt

    +1 t

    x1etdt

    limt

    tx1et1t2

    = limt

    tx+1

    et= 0,

    1

    1t2dt < . pi

    +1 t

    x1etdt (x) x 1) x > 1 : pipi

    +1 t

    x1etdt pi pipi ).

    10 t

    x1etdt . t = 1s 1

    0tx1etdt =

    +1

    ts1e1/sds.

    pi

    lims

    ts1e1/s

    sx1= 1,

    1 s

    x1ds < x > 0 pi

    10 t

    x1etdt (x) x > 1.

    4.2.18

  • Riemann

    .

    JJ II

    J I

    261 pi 273

    pi :

    B(x, y) = 10tx1(1 t)y1dt =

    c0tx1(1 t)y1dt +

    1ctx1(1 t)y1dt,

    0 < c < 1.pi f (t) = tx1(1 t)y1

    limt t

    1x f (t) = limt(1 t)

    y1 = 1,

    limt(1 t)

    1yf (t) = limt t

    x1 = 1.

    pi pi c0 t

    x1(1t)y1dt 0 < 1 x < 1 1

    ctx1(1 t)y1dt 0 3. pi pi x = 1 x = 3. x = 1 n=1 n+1n2+1 (1)n, pi pi ( an = n+1n2+1 ) . x = 3 n=1 n+1n2+1 pipi.pi n=1 n+1n2+1 (x 2)n [1,3) R = 312 = 1.

    4.2.22

  • Riemann

    .

    JJ II

    J I

    265 pi 273

    pi :

    limn

    n

    3n2n+4 |x

    n | = 32 |x |.

    pi |x | < 23 23 < x < 23 pi x > 23 x < 23 . pi pipi pi x = 23 x = 23 . x = 23

    n=1

    18 (1)n. pi pi-

    . x = 23

    n=1

    18 . pi pi.

    pi n=1 3n2n+4 xn ( 23 , 23 ) R = 23 .

    4.2.23

  • Riemann

    .

    JJ II

    J I

    266 pi 273

    pi :

    limn

    xn+1

    (n+1)!xn

    n!

    = limn

    x

    n + 1 = 0,

    x , 0 n=1 xnn! = 0 x = 0. x R (,+) R = +.

    4.2.24

  • Riemann

    .

    JJ II

    J I

    267 pi 273

    pi : x , 1

    limn

    nnn(x 1)n = lim

    nn|x 1| = +.

    x = 1 n=1 nn(x 1)n = 0 . pi - [1,1] R = 0.

    4.2.25

  • Riemann

    .

    JJ II

    J I

    268 pi 273

    pi : fn(x) = (1)n+1 xnn x [0, a] a < 1,

    |fn(x)| = |(1)n+1 xn

    n| = x

    n

    n a

    n

    n.

    n=1 ann 0 a < 1 pi x (1,1) R = 1.

    4.2.26

  • Riemann

    .

    JJ II

    J I

    269 pi 273

    pi : fn(x) = xn ln(x)n x [0,1] . -

    x [0,1] |fn(x)| = fn(x) fn(x) xm = 1ne fn(xm) = 1n2e

    |fn(x)| = xn ln(x)n

    1n2e

    .

    n=1 1n2e pi x [0,1].

    4.2.27

  • Riemann

    .

    JJ II

    J I

    270 pi 273

    pi :

    limn

    nanxn = lim

    na|x | = a|x |.

    pi pipi a|x | < 1 |x | < 1a . - ( 1a , 1a ) R = 1a .

    4.2.28

  • Riemann

    .

    JJ II

    J I

    271 pi 273

    pi : f : (R, R) R pi f (x) = n=0 anxn. f (x) = n=0 nanxn. pi pi

    2f (x) + f (x) = 0 2e2x f (x) + e2x f (x) = 0 (e2x )f (x) + e2x f (x) = 0 (e2x f (x)) = 0 e2x f (x) = c f (x) = ce2x .

    ex = n=0 xnn! e2x = n=0 (2x)nn! f (x) = cn=0 (2)nn! xn. pin=0

    an =n=0

    (c(2)nn!

    )xn.

    pi an = c (2)n

    n! .

    4.2.29

  • Riemann

    .

    JJ II

    J I

    272 pi 273

    pi : n=0 anxn R, |x | < R. pi n=0 anx2n = n=0 an(x2)n pi |x2| < R |x | < R. n=0 anx2n

    R.

    4.2.30

  • Riemann

    .

    JJ II

    J I

    273 pi 273

    pi : pi

    limn

    (n + 1)axn+1

    naxn= lim

    n((n + 1)

    n)ax = x.

    R = 1. x = 1 n=1 na(1)n pi a < 0 . x = 1 n=1 na pi a < 1.pi a < 1 [1,1], 1 a < 0 [1,1) a 0 (1,1).

    4.2.31

    A'oristo Olokl'hrwmaStoiqe'ia Jewr'iacAsk'hseic

    Olokl'hrwma RiemannStoiqe'ia Jewr'iacAsk'hseic

    Seir'ecStoiqe'ia Jewr'iacAsk'hseic

    Genikeum'ena oloklhr'wmataStoiqe'ia Jewr'iacAsk'hseic