Shepin e v Lekcii Po Analizu v Sunc Integraly

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Transcript of Shepin e v Lekcii Po Analizu v Sunc Integraly

  • ..

    . .

    2011

  • 1 . . . . . . . . . . . . . . . . . . . . . . . . . 22 . . . . . . . . . . . . . . . . . . . . . . 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 . . . . . . . . . . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . 216 . . . . . . . . . . . . . . . . 257 . . . . . . . . . . . . . . . . . . . . . 298 . . . . . . . . . . . . . . . . . . . . . . . 329 - . . . . . . . . . . . . . . . 36

    1

  • 1

    bRa

    f(x) dx, , -

    - , - x [a; b]. - f(x), x, - dx, x. - - dx ( x) . , "", - , - dx. -, [a; b] . , :

    (1.1)bZ

    a

    dx = b a

    - - badx . cdx, - cdx badx =c(b a). - :

    (1.2)bZ

    a

    c dx = c(b a)

    , , - :

    bZa

    cf(x) dx = c

    bZa

    f(x) dx(1.3)

    bZa

    f(x) dx+

    bZa

    g(x) dx =

    bZa

    (f(x) + g(x)) dx(1.4)

    ,

    (1.5)bZ

    a

    f(x) dx+

    cZb

    f(x) dx =

    cZa

    f(x) dx;

    , , 1R0

    xn dx. 1R0

    x dx. -:

    1Z0

    x dx =

    12Z

    0

    x dx+

    1Z12

    x dx:

    2

  • x! x+ 12 1R12

    x dx -

    12R0

    (x+ 12 ) dx. ,

    :

    (1.6)bZ

    a

    f(x) dx =

    b+cZa+c

    f(x c) dx;

    . - , - : f(x)dx x - x+ c , f(x)dx. . - :

    (1.7)1Z

    12

    x dx =

    12Z

    0

    (x+1

    2) dx =

    dZ12

    x+

    1Z12

    x dx =1

    2+

    1Z12

    x dx

    (1.8)1Z

    0

    x dx = 2

    12Z

    0

    x dx+1

    2

    - , . - x ! kx c k. , y = kx

    bRa

    f(x) dx. x a b, y ka,

    kb kbRka

    f(y) dy. , ,

    y x. ? . dy dx. -, k , k. - ! dx dy = kdx. :

    (1.9)bZ

    a

    f(x) dx = k

    kbZka

    f(x=k) dx:

    -! .

    3

  • , -

    1R0

    x dx. , y = 2x

    1Z0

    x dx = 4

    12Z

    0

    x dx:

    (1.8) :1R0

    x dx = 12 .

    1R0

    xn dx

    , 1R0

    x dx. -

    y = ax aR0

    xn dx,

    1R0

    xn dx. bRa

    f(x) dx =bR0

    f(x) dxaR0

    f(x) dx,

    bRa

    xn dx. -

    bRa

    f(x) dx , .

    . , , . :

    (1.10) f(x) g(x) x 2 [a; b], bZ

    a

    f(x) dx bZ

    a

    f(x) dx

    bRa

    f(x) dx

    . [a; b] , -

    a = x0 < x1 < < xn1 < xn = b, [a; b]

    bZa

    f(t) dt =nX

    k=1

    xkZxk1

    f(t) dt

    (xk xk1)min f [xk1; xk] xkZ

    xk1

    f(t) dt (xk xk1)max f [xk1; xk]

    4

  • :

    nP

    k=1

    (xk xk1)max f [xk1; xk] -

    nP

    k=1

    (xk xk1)min f [xk1; xk]. - jf(b) f(a)jmax(xk xk1). , .

    -. : , - . .

    1. F (x; dx) G(x; dx) , - .

    . , - . , " > 0 -

    bZa

    F (x; dx) bZ

    a

    G(x; dx) + "

    q F (x) qG(x), -

    I(F ) =

    bZa

    F (x) qbZ

    a

    G(x) = qI(G)

    q < 1 + "=I(G) . . , jI(F ) I(G)j < " ". .

    - , - - . , - (??) xn,

    dxn = (x+ dx)n xn = nxn1dx+ C2nxn2(dx)2 + = nxn1dx

    , .

    5

  • . - . - - . , , . - .

    , OPQ (O ) R = OP = OQ PQ, .

    ABC AB R AC PQ. - ABC OPQ. BC PQ . - . () () A (. O), ( ). () - ABC, () OPQ. - , - . .

    , - f(x) [a; b] - . dx , - x. - ds f(x)dx, dx f(x) (< df(x)dx). ,

    bRa

    f(x) dx

    .

    , bRa

    xn dx y = xn

    [a; b]. dxn+1

    n+1 xndx dx , bRa

    xn dx 1n+1bRa

    dxn+1 =

    bn+1an+1n+1 .

    . -

    . -

    6

  • . , x x(t).

    t1Rt0

    y(t) dx(t) -

    . f(t) t df(t) -

    dt t. dt , , -. -

    (1.11)bZ

    a

    df(x) = f(b) f(a)

    ., s(t), - h. bRa

    s(h) dh , ,

    a b. , , dh h h- .

    .

    1. .

    2. R jdf(x)j.

    3. Rg(x)df(x).

    4. R(g(x)df(x) f(x)dg(x).

    7

  • 2 . F (z) - f(z) -

    Rf(z)dz.

    bRa

    f(z) dz -

    F (b) F (a) . , , , . , f(z) , .

    (2.1)bZ

    a

    f(x) dx = F (b) F (a)

    b < a. -

    (2.2)bZ

    a

    f(x) dx = aZb

    f(x) dx;

    .

    -:

    (2.3)bZ

    a

    f(t)dt+

    cZb

    f(t)dt =

    cZa

    f(t)dt

    a; b; c.

    .

    2.1 ( ). [a; b] f(x) , f()(b a) S f(x) [a; b].

    . , f(x) > S=(b a), f(x) S=(ba)(ba) S, S, -. , f(x) < S=(b a) . , - , . .

    1. (a; b) f(x) - .

    8

  • . f(x). - x0 2 (a; b) . F (x) - f(x) [x0; x], x > x0 f(x) [x; x0], x < x0.

    f(x) - (a; b). x0 2 (a; b) F (x) , f(x), - x0 x, x > x0 , , x < x0. , F 0(x) = f(x).

    [x; x+nx], xn , f(xn) ,

    f(xn) =F (x+nx) F (x)

    nx

    nx , xn x. f(x) f(x). -, F 0(x) = f(x).

    f(x) f+(x) =12 (f(x) + jf(x)j) f+(x) = 12 (f(x) + jf(x)j) . f(x), f(x). F(x). F+(x)F(x) f(x).

    . . , [x] . - , - , , - .

    - , , - , - - . , -. - [a; b] x1; : : : ; xn , . x0 = a xn+1 = b. (xk; xk+1) Fk(x)+ck ck, fFk(x) + ckg , - .

    . f(x) ,

    F (x). bRa

    f(x) dx -

    9

  • (b) = F (b) F (a), , , , 0(b) = F 0(b) = f(b).

    2 ( ). : [t0; t1]! [0; 1] (t0) = 0 (t1) = 1 f(t) f((t)) 0(t) [t0; t1].

    (2.4)t1Z

    t0

    f((t)) 0(t) dt =

    1Z0

    f() d;

    . F 0() = f(), - F ((t)) f((t)) 0(t). - (2.4) F (1) F (0), F ((t1)) F ((t0)),

    . -

    bZa

    cf(x) dx = c

    bZa

    f(x) dx(2.5)

    bZa

    (f1(t) + f2(t)) dt =

    bZa

    f1(t) t+

    bZa

    f2(t) dt(2.6)

    F 0(t) = f(t), (cF (t))0 = cf(t), . F 0i (t) = fi(t), (F1(t) +F2(t))0 = f1(t) + f2(t), .

    t 2 [a; b] - f1(t) f2(t), .

    (2.7)bZ

    a

    f1(t) dt bZ

    a

    f2(t) dt

    . b = a . , f1(b) , f2(b), b > a, (2.7) .

    . f(x) - x - n ! 0, - dx x, - df(x)

    10

  • ff(x+n) f(x)g, dx f 0(x). df(x) = f(x+dx)f(x) -

    (2.8) df(x) = f 0(x)dx;

    . Rf(t)dg(t)

    Rf(t)g0(t)dt.

    3 ( .). f(x) g(x) -,

    (2.9)bZ

    a

    f(t) dg(t) = f(b)g(b) f(a)g(a)bZ

    a

    g(t) df(t)

    . -, f 0(x)g(x) f(x)g0(x) - -. a b. f(b)g0(b), (f 0(b)g(b) + f(b)g0(b)) g(b)f 0(a). - . b = a .

    .

    1.R

    dxpx2x

    2.Rxexdx

    3.Rx5e2x

    2

    dx

    4.Rx cosxdx

    5.Rx3 lnxdx

    6.Rex sinxdx

    7.Rlnxdx

    8.Rarctg xdx

    9.Rsin4 xdx

    10.R

    dx1+x3

    11.

    1R0

    sin ktt dt

    0k

    11

  • 3 . -. , x! x. - (x), x - -, - :

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

    (1) = 1, , n (n) = (n 1)!.

    , - :

    (3.2) (x) = limn!1

    nxn!

    x(x+ 1) : : : (x+ n)

    . - (x) = ln(x) ( -):

    (x) = lnx:

    . , ,

    . f ,

    x < y < z - :

    (3.3)f(y) f(x)

    y x f(z) f(y)

    z y ax + b, -

    a.

    3.1. ( ) .

    . - [a; b] .

    3.2. f(x) f(a)f(x)ax x a

    . x > y > a, f(x) f(y) f(x) (f(x) f(a))yxxa . f(x) f(a),

    f(y) f(a) (f(x) f(a))1 +

    y xx a

    = (f(x) f(a))y a

    x a;

    y a - .

    12

  • .

    3.3. f 00(x) 0 [a; b], -

    f 0(a) f(a) f(b)a b f

    0(b)

    . f 0(x)0 0, f 0(x) - . g(x) = (f(x)f(a))f 0(a)(xa). f 0(x) f(a) . , g(x) g(b) g(a) = 0. . .

    3.4. f(x) [a; b],

    f 0(a) f(a) f(b)a b f

    0(b)

    . f 0(b) = limn!1

    f(b)f(xn)bxn ,

    xn = b 1n , f(b)f(xn)bxn - f(a)f(b)ab 3.2, . .

    1. f(x), [a; b], , f 00(x) 0 x .

    . f 00(t) 0. x 0, - (x+1) = x(x) 1.

    14

  • . G(x) = x+12

    x2

    . G(x+

    1) = x+22

    x+12

    =

    x2 + 1

    x+12

    = x2

    x2

    x+12

    = x2G(x). -

    G(x)2x -. G(x) , - , , G(x)=G(1) , G(x)2

    x

    G(1)21 = (x). x 2x

    (3.4) (2x) =22x1(x+ 0:5)(x)

    (0:5)

    .

    1. .

    2. (0) (1)?3. lim

    x!0(x n)(x n)

    4. 1p +1q = 1 xy x

    p

    p +yq

    q

    5. , 1Qk=1

    kx+k

    k+1k

    x= (x+ 1)

    6. , xk = kxk1 k, xk =(x+1)

    (xk+1)

    7. , xk+m = xk(x k)m.8. , 1x2 -

    .

    9. , 1x

    10. . lnx .

    11. , -.

    12. , .

    13. xx12 ex e(x) (x), (x)

    (x) = 1 x+ 12 ln 1 + 1x

    15

  • 4 . -:

    1 (Euler). x 0 (x) =1R0

    tx1et dt

    , -

    . x = 1 1R0

    et dt = et j10 =