Конспект лекций по математическому анализу. Шерстнев А.Н. - 213 стр.

| NEPRERYWNOE PRODOLVENIE PODYNTEGRALXNOJ FUNKCII NA PRQMOUGOLX-
NIK f(x; )j a  x  b; 0    1g (!!). iZ P. 2
               Zb          2               Zb             Zb
    lim p 1      expf, x 2 g dx = lim   f (x; ) dx = f (x; 0)dx = 0:
   !0+ 2    a      2           !0+ a               a
aNALOGI^NO RAWEN NUL@ PREDEL INTEGRALA PRI a < b < 0. pUSTX TEPERX
a = 0 < b. pODYNTEGRALXNAQ FUNKCIQ W \TOM SLU^AE NE PRODOLVAETSQ PO
NEPRERYWNOSTI NA PRQMOUGOLXNIK [0; b]  [0; 1]. dELAQ W INTEGRALE ZAMENU
t = x=, IMEEM
                    Z                                Z
        lim p 1 b expf, x22 g dx = lim p1 b= expf, t2 g dt
       !0+ 2 0         2        !0+ Z 2 0            2
                                           + 1         2
                                  = p1         expf, t2 g dt:
                                      2 0
s U^ETOM 132.10 IMEEM OKON^ATELXNO
                                       8
               1 Z b        x 2        < 0; ESLI 0 62 [a; b],
                                       >
         lim p       expf, 22
                                g dx = > 1=2; ESLI a = 0 ILI b = 0,
        !0+ 2 a                     : 1; ESLI 0 2 (a; b).

   x134. dIFFERENCIROWANIE SOBSTWENNYH INTEGRALOW
   1.  pUSTX = U  [c; d], GDE U OTKRYTO W R, I f; @f          @x NEPRERYWNY NA
  . tOGDA
                   d Z d f (x; y) dy = Z d @f (x; y) dy (x 2 U ):
                  dx c                  c @x
                        Zd
  pOLOVIM F (x) = c f (x; y) dy (x 2 U ). pUSTX [a; b]  U I a < x < b. pO
FORMULE lAGRANVA IMEEM DLQ MALYH h
    F (x + h) , F (x) = Z d f (x + h; y) , f (x; y) dy = Z d @f (x + h; y) dy;
            h               c            h                  c @x
ZDESX 0 <  < 1. pRIMENQQ 133.2 K 0 = [a; b]; 00 = [c; d], IMEEM
             F 0(x) = hlim  1 [F (x + h) , F (x)] = Z d @f (x; y) dy: >
                        !0 h                          c @x

                                     213