.    .
 


              .      ,         .       .       ,    ,     .





 

.    .





:

1. .

2.  .

3.    .

4.     .

5.  ,   .

6.      .

7.       .

8.    ,   .

9.     .

10.   .

11.        .

12.       ,    .

13)    .



.    .

1. .

   .

      .

   .

    ,  .

   ,    -  S.

      -  t.

    -  -    ().

 t ()   .  S (t) (     )   .    S (t) -  .

    ,  .

  ,     (  ).

  ,     (  ).








S (t)  , ,  .

t  , , .

 ,   ( t = 0 S = 0)

     .     ,   .

 S (t)     .

    .  ,    .

   ,     .

     :

v  = (S (t + ? t)  S (t)) / ? t

   ? t   ,      .

 (? t ? 0) ? t   ,     .

       t -  ,       ? t ? 0

v (t) = lim    (S (t + ? t) - S (t)) / ? t

? t?0

     S(t).

v (t) = S ' (t)  v (t) = ? (t)

    .

,        .

v (t) = S ' (t)

       S (t)    v (t)

      ,      .

   t   x,  S   f.

f ' (x1) = lim (f (x1 + ?x)  f (x1)) / ?x      ? x ? 0

      ,     .    ,   ,  .

 : F (x)

    ,       .

:

 F (x)     f (x),   ;    x     : F ' (x) = f (x)

    ,   ,   .

2.  .

  ,      .

? f (x) dx = F (x) + C

?   .

f (x)  .

dx    x.

F(x)  .

C  ( ).

 .

1) ,    ,         .

? c f (x) dx = c ? f (x) dx = ? f (x) d cx

2)     ,      .

c ? f (x) d /c = 1/c ? f (x) d cx

c     ().

3)  ,   .

? (f (x) + g (x)) dx = ? f (x) dx + ? g (x) dx

4)  ,   .

? (f (x) - g (x)) dx = ? f (x) dx - ? g (x) dx

5)     ,    .

? f (x) dx = ? d F (x)

6)    - ,    .

? d f (x) = ? f ' (x) dx

7)         0,     ,     ,    C.

? f (x) dx = F (x) + C

C   ().

8)       .     .

? f (x) dx = ? f (x) d(x + )

3.      (  ).

1) ?  dx = Cx + C1

2) ? 1 dx = x + C

3) ?  ? =  ??? / (n + 1) + C;   n?1

4) ? ?x = (2/3) ?x?

5)  ? (1/x) dx = ? dx /x = ln |x| + C

6) ? e * d = e * + C

7) ? cos x dx = sin x + C

8) ? sin x dx = - cos x + C

9) ? dx / cos ? x = tg x + C

10) ? dx / sin ? x  = - ctg x

11) ?  * d  = (a * / ln) + C

12) ? dx / (1 + x ?) = arctg x + C

13) ? dx / ?(1   ?) = arcsin x + C

14) ? dx / (x ln?) = log?  x + C

15) ? dx / ?x = 2?x + C

16) ? dx / ?(x ?  a ?) = ln|x + ?(x ?  a ?)|+ C

17) ? dx / (x ? - a ?) = (1/2a) ln |((x - )/( + ))| + C

18) ? dx / ( ? -  ?) = (1/2a) ln |((x + )/( - ))| + C

19) ? dx / x ? = - 1/x + C

20) ? dx / ( ? + x ?) = (1/a) arctg (x/a) + C = - (1/a) arctg (x/a) + C

21) ? dx / ?(  ? - x ?) = arcsin (x/a) + C = - arccos (x/a) + C

22) ? sh  dx = ch  dx + C

23) ? ch  dx = sh  dx + C

24) ? dx / ch ? x = th x + C

25) ? dx / sh ? x = - cth x + C

4.     .

1))    :

? x ? = x ???/ (n + 1) + C

   (     )       .

     .

 -      .

 -      .

   -      .

2))       ,    .

1) ,   ,          .

?  f (x) dx = c ? f (x) dx = ? f (x) d (cx)

2)           .

? c f (x) d (/) = 1/c ? f (x) d cx = ? f (x)/c d cx = ? c f (x) d (x/c)

3)       .     .

? f (x) dx = f (x) d(x + c)

 ,    - ,    .     .

 .

? x ? dx =

  .    .

= x ???/ (2 + 1) + C = x ?/ 3 + C

:    x ?/ 3 + C

 .

? 1 / x ? dx =

 ,   .  ,    .

? x ?? dx =

  .    .

= x ????/(- 3 + 1) + C = x ??/(- 2) + C

 ,    .

= ((- 1/2) 1/x ?) + C = - 1/(2 x ?) + C

:  - 1/(2 x ?) + C

  .

? ?x dx =

 ,   .    .

= x ??? dx =

  .    .

=  x ?????/ (1/2 + 1) + C =   ???/ (3/2) + C =

   .

=  2 ?(x ?) / 3 + C = 2 x ?x / 3 + C

:   2 x ?x / 3 + C

5.  ,   .

   ,      .

     ,    ,     .

   , ? ,       (  ).








? f (x) dx = F (x) + C

    ,  . ,    y = F (x) + C     y = F (x)     y.

  C   ,       .

6.      .

 .

? x ? dx =

  .    .

= x ???/ (3 + 1) + C = x ? / 4 + C

:  x ? / 4 + C

 .

? ??x dx =

   .    .

= ? x ??? dx =

  .    .

= x ?????/ (1/3 + 1) + C = x ???/ (4/3) +  = (3/4) x ??? + C =

   .

= (3/4) ??x? + C = (3/4) x ??x + C

: (3/4) x ??x + C

 .

? (1/ ?x) dx =

   .  ,    .      .

= ? x ???? dx =

  .    .

=  x ??????/(- 1/2 + 1) + C = x ???/ (1/2) + C =

   .    .

= 2 ?x + C

:   2 ?x + C

 .

? (1 / x ?) dx =

 ,   .   ,    .

= ? x ?? dx =

  .    .

= ? x ????/(- 2 + 1) + C = x ??/(- 1) + C =

 ,    .   .

=  - 1/ x + C

:   - 1/ x + C

 .

? dx / ??x = ? (1 / ??x) dx =

 ,   .

  ,    .    .

= ?  ???? dx =

  .    .

= ? x ??????/(- 1/3 + 1) + C = x ???/ (2/3) + C =

          .   .

(3/2) ??x? + C

:   (3/2) ??x? + C

     ,       ()      .

? dx / f (x) = ? (1/ f (x)) dx

 .

? dx / ??= ? (1/ ?x?) dx =

 ,    .    .

= ? x ???? dx =

  .    .

= x ??????/(- 3/2 + 1) + C =  x ????/ (- 1/2) + C

      .

=  - 2/ ? + C

:   - 2 / ?x + C

 .

? 2x ? dx =

     .

2 ?  ? dx = 2 x ? / 5 + C = (2/5) x ? + C

:  (2/5) x ? + C

 .

? ?(2x) dx =

       .      ,           ( 2)

= 1/2 ? ?(2x) d 2x =

     .

= 1/2 ? 2/3 ?(2x) ? + C = 1/3 ?(8x ?) + C = 2/3 ?(2x ?) + C

:   2/3 ?(2x ?) + C

 .

? (x + 1) ? dx =

       ,     .

     ,       .       .   -  ,   .    .

= ? (x + 1) ? d(x + 1) =

 .  .

= ( + 1) ???/(3 + 1) + C = (x + 1) ?/ 4 + C

:   (x + 1) ?/4 + C

 .

? ?(x + 5) dx =

      .

     .     .

= ? ? (x + 5) d (x + 5) =

 .      .

= 2/3 ? (x + 5) ? + C

:    2/3 ? (x + 5) ? + C

 .

? dx / ?(3 - x) = ? (1 / ?(3 - x)) dx =

 .   .    .

= ? (3 - x) ???? dx =

      .      ,            .     .

= - ? (3 - x) ???? d(- x + 3) = - ? (3 - x) ???? d(3 - x) =

    .

= - (3 - x) ?????? /(- 1/2 + 1) + C = - (3 - x) ???/(1/2) + C =

        .

= - 2 ?(3 - x) + C

:  - 2 ?(3 - x) + C

 .

? ( 3 - 4x) ? dx =

      .             .

= - 1/4 ? (3 - 4x) ? d(- 4x + 3) = - 1/4 ? (3 - 4x) ? d(3 - 4x) =

 .   .

= - 1/4 (3 - 4x) ???/(5 + 1) + C = - 1/4 (3 - 4x) ?/ 6 + C = - 1/4 (3 - 4x) ?/ 24 + C

:    - 1/4 (3 - 4x) ? / 24 + C

 .

? (5x + 7) ? dx =

      .      5       7.

= 1/5 ? (5 + 7) ? d(5 + 7) =

 .  .

= 1/5 (5x + 7) ??? / (8 + 1) + C = 1/5 (5x + 7) ? / 9 + C = (5x + 7) ? / 45 + C

:   (5x + 7) ? / 45 + C

 .

? (4 - x /2) ?dx =

      .      -2       4.

= - 2 ? (4 - x/2) ? d(- x/2 + 4) = - 2 ? (4 - x/2) ? d(4 - x/2) =

 .   .

= - 2 (4 - x /2) ???/(7 + 1) + C = - 2 (4 - x /2) ? / 8 + C = - (4 - x /2) ? /4 + C

:    - (4 - x /2) ? / 4 + C

7.      :

1)     .

? (f (x) + g (x)) dx = ? f (x) dx + ? g (x) dx

2)     .

? (f (x) - g (x)) dx = ? f (x) dx - ? g (x) dx

 .

? (3x ? + 5x) dx =

    .

? 3x ? dx + ? 5x dx =

    .

= 3 ? x? dx + 5 ? x dx = 3x ? /3 + 5x ? /2 + C = x ? + (5/2) x ? + C

:   x ? + (5/2) x ? + C

    ,     .

 .

? (?x - x ? /2) dx =

    .

= ? ?x dx - 1/2 ? x ? dx = (2/3) ?x? - 1/2 (x ? / 3) + C = (2/3) ?x? - x ? / 6 + C

:   (2/3) ?x? - x ? / 6 + C

 .

? (4 x ? + 5 ??x) dx =

    .

? 4 x ? dx + 5 ?  ??x dx =

           .

= 4 ? x ? dx + 5 ? x ??? dx =

  .    .

4 x ??? / (3 + 1) + 5 x ????? / (1/3 + 1) + C = 4 x ? / 4 + 5 x ??? / 4/3 + C =

     .

= x ? + (15/4) ??x? + C

:   x ? + (15/4) ??x? + C

 .

? (5x ? - 2x) dx =




  .


   .

   ,     (https://www.litres.ru/book/vasilisa-volkonskaya/integraly-bazovyy-kurs-dlya-nachinauschih-74196172/)  .

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