  
 


           ,   .      ,      ,   .       .       ,    ,     .





 

  



:

1)    .

2)     .

3)      .

4)   .

4. 2       ( ).

4.1      .

5)    .

5. 1      .            .

5. 2     .

        ,      .

5. 3    .

       .

5. 4    .

       .



1)    .

  (  ).

1)  ' = 0

 -  .

 .

2) (x ?) ' = n x ???

   .

1. x ' = 1

2. c x ' = c

c  c1 -  .

3. (x ?) ' = 2x

4. (x ?) ' = 3x ?

5. (1/ ?x) ' = - 1/2 ? 1 / ?x ?

6. (??x) ' = 1 / (3 ??x ?)

7. (?x) ' = 1 / (2 ?x)

8. (1 / x) ' = - 1 / x ?

9. (1 / ??x) ' = - 1 / (3 ??x ?)

  .

3) (ln x) ' = 1 / x

4) (e *) ' = e *

5) (cos x) ' = - sinx

6) (sin x) ' = cos x

7) (tg x) ' = 1 / cos ? x

8) (ctg x) ' = - 1 / sin ? x

9)  (a *) ' = a * ln a

a > 0

a ? 1

x > 0

10) (arctg x) ' = 1 / (1 + x ?)

11) (arctg x /a) ' = a / (a ? + x ?)

12) (arcctg x) ' = - 1 / (1 + x ?)

13) (arcctg x /a) ' = - a / (a ? + x?)

14) (arcsin x) ' = 1 / ?(1 - x ?)

15) (arcsin x/a) ' = 1 / ?(a ? - x ?)

16) (arccos x) ' = - 1 / ?(1 - x ?)

17) (arccos x/a) ' = - 1 / ?(a ? - x ?)

 .

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

2)     .

 .

? dx / cos x =

     cos x.

= ? cos x dx / cos ? x =

   ,       .

= ? cos x dx / (1 - sin ? x) =

   ,   .

? d sin x / (1 - sin ? x) =

  .

 sin x = t, :

= ? dt / (1 - t ?) =

  .    .

1/2 ln |(t + 1) / (t - 1)| + C =

   .

= 1/2 ln |(sin x + 1) / (sin x - 1)| + C

: 1/2 ln |(sin x + 1) / (sin x - 1)| + C

 .

? dx / sin x =

     sin x,     .

= ? sin x dx / sin ? x =

   ,   .

        .

= - ? d cos x / (1 - cos ? x) =

 ,    .

= - 1/2 ln |(cos x + 1) / (cos x - 1)| + C

: - 1/2 ln |(cos x + 1) / (cos x - 1)| + C

 .

? (cos ? x / sin ? x) dx =

    ,   .

= ? cos ? x d sin x / sin ? x =

       .

= ? (1 - sin ? x) d sinx / sin ? x =

  .

 sin x = t, 

= ? (1 - t ?) dt / t ? =

    .

= ? dt / t ? - ? t ? d t / t ? = ? dt / t ? - ? dt =

     .

= - 1 / t - t + C =

  .

= 1 / sin x - sin x + C

: 1 / sin x - sin x + C

 .

? tg ? x dx =

        .

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

    .

= ? dx / cos ? x - ? dx =

  .

= tg x - x + C

:  tg x - x + C

 .

? ctg ? x dx =

        .

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

    .

= ? dx / sin ? x - ? dx =

 ,    .

= - ctg x - x + C

: - ctg x - x + C

 .

? tg ? x dx =

     .

=? (sin ? x/ cos ? x) dx =

    ,   .

= ? (sin ? x d (- cos x) / cos ? x) dx =

              .

= - ? (1 - cos ? x) d cos x / cos ? x) dx =

  .

 cos x = t

= - ? (1 - t ?) dt / t ? =

    .

= - ? (1 / t ? - t ? / t ?) dt = - ? 1 / t ? - 1 / t) dt =

    .

- 1 / t ? dt + ? dt / t = - t ?? dt + ? dt / t =

   .

- t ?? / (- 2) + ln |t|+ C = 1 / 2t ? + ln |t| + C =

  .

= 1 / 2cos ? x + ln |cos x| + C

: 1 / 2cos ? x + ln |cos x| + C

 .

? ctg ? x dx =

     .

= ? ctg ? x ctg x dx =

       .

= ? (1 / sin ? x - 1) ctg x dx =

 .

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

    .

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

   ,   .

= ? ctg x d (- ctg x) - ? cos  / sin x dx = - ? ctg x d ctg x - ? d sin x / sin x =

 ,    .

= - ctg ? x /2 - ln |sin x| + C

: - ctg ? x /2 - ln |sin x|+ C

  .

 -    .

sec x = 1 / cos x

 -     .

cosec x = 1 / sin x

 8.

? tg x sec x dx =

   .

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

    ,   .

= ? tg ? x d tg x =

 .

    .

(tg ? x) / 3 + C

: 1/3 tg ? x + C

 .

? ctg ? x cosec ? x dx =

   .

? ctg x ? 1 / sin ? x dx =

    .

= ? ctg ? x d ctg x =

 .

    .

= - (ctg ? x) / 3 + C

: - (ctg ? x) / 3 + C

.

? tg x cosec ? x dx =

   .

= ? tg x ? 1 / sin ? x dx =

    ,   .

= ? tg x d (- ctg x) = - ? tg x d ctg x =

   ,   .

= - ? 1 / ctg x d ctg x = - ? d ctg x / ctg x =

 .

    .

= - ln |ctg x| + C

: - ln |ctg x| + C

 10.

? ctg ? x sec ? x dx =

   .

= ? ctg x ? 1 / cos ? x dx =

    .

  .

= ? ctg ? x d tg x =

      .

= ? 1/ tg ? x d tg x = ? tg ?? x d tg x =

      .    .

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

: - 1 / 2 tg ? x + C

 11.

? tg ? x sec ? x dx =

   .

= ? tg ? x ? 1 / cos ? x dx =

  -.

= ? tg ? x / cos ? x ? 1 / cos ? x dx =

    .

  .

= ? tg ? x / cos ? x d tg x =

          .

= ? tg ? x ? 1 / cos ? x d tg x = ? tg ? x (1 + tg ? x) d tg x =

  .

 tg x = t

= ? t ? (1 + t ?) dt =

 .

= ? (t ? + t ?) dt =

    .

= ? t ? dt + ? t ? dt =

      .

= t ?/ 5 + t ? / 7 + C =

    .

= tg ? x / 5 + tg ? x / 7 + C =

: tg ? x/ 5 + tg ? x / 7 + C

 12.

? sin ? x dx =

         .

? 1/2 (1 - cos 2x) dx = ? (1/2 - 1/2 cos 2x) dx =

    .

= ? 1/2 dx - 1/2 ?cos 2x dx =

      .

= ?1/2 dx  1/4 ? cos 2x d2x =

 .

    .

= 1/2 x - 1/4 sin 2x + C

: 1/2 x - 1/4 sin 2x + C

 13.

? cos ? x dx =

          .

= ? 1/2 (1 + cos 2x) dx = (1/2 + 1/2 cos 2x) dx =

    .

= ? 1/2 dx + 1/2 ? cos 2x dx =

      .

? 1/2 dx + 1/4 ? cos 2x d2x =

 .

    .

= 1/2 x + 1/4 sin 2 + C

:  1/2x + 1/4 sin 2 + C

 14.

? sin ? x dx =

     .

= 1/4 (3 sin x - sin 3x) dx =

 .

  .

= 3/4 ? sin x dx - 1/4 ? sin 3x dx =

      .

= 3/4 ? sin x dx - 1 / 12 ? sin 3x d 3x =

 .

   .

= - 3/4 cos  + 1/12 cos 3 + C

: - 3/4 cos  + 1 / 12 cos 3 + C

 15.

? cos ? x dx =

     .

= ? 1/4 (3 cos x + cos 3x) dx =

     .

= ? 3/4 cos x dx + 1/4 ? cos 3x dx =

      .

= ? 3/4 cos x dx + 1/12 ? cos 3x d 3x =

 .

    .

= 3/4 sin x + 1/12 sin 3x + C

:  3/4 sin x + 1/12 sin 3x + C

 16.

? sin x cos 2x dx =

   .

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

    ,     .

1/2 ? (sin 3x - sin x) dx =

    .

= 1/2 ? sin 3x dx - 1/2 ? sin x dx =

      3.

= - 1/6 ? sin 3x d 3x + 1/2 ? sin x dx =

 .

    .

= - 1/6 cos 3 + 1/2 cos x + C

: - 1/6 cos 3 + 1/2 cos x + C

 17.

? sin 2x sin 3x dx =

   .

= ? 1/2 (cos (2x - 3x) - cos (2x + 3x)) dx = 1/2 ? (cos (- x) - cos 5x) dx =

    ,   .

= 1/2 ? (cos x - cos 5x) dx =

    .

= 1/2 ?cos x dx - 1/2 ? cos 5x dx =

      .

= 1/2 ? cos x dx - 1/10 ? cos 5x d 5x =

 .

    .

= 1/2 sinx - 1/10 sin 5x + C

:  1/2 sinx - 1 / 10 sin 5x + C

 18.

? cos 3x cos x dx =

   .

= ? 1/2 (cos (3x - x) + cos (3x + x)) dx = 1/2 ? (cos 2x + cos 4x) dx =

    .

= 1/2 ? cos 2x dx + 1/2 ? cos 4x dx =

   .

= 1/4 ? cos 2x d2x + 1/8 ? cos 4x d 4x =

 .

    .

= 1/4 sin 2x + 1/8 sin 4x + C

: 1/4 sin 2x + 1/8 sin 4x + C

 19.

? dx / sin ? 2x =

   .

= ? dx / (2 sin x cos x) ? = 1/2 ? ? ? dx /sin ? x cos ? x = 1/8 ? dx / sin ? x cos ? x =

     cos ? .

1/8 ? cos ? x dx / sin ? x cos ? x =

      .

= 1/8 ? ctg ? x ? 1/cos ? x ? 1 / cos ? x  dx =

    .

  .

1/8 ? ctg ? x ? (1 /cos ? x) ?  d tg x=

      .

= 1/8 ? ctg ? x ? (1 + tg ? x) ? d tg x =

  .

 tg x = t

= 1/8 ? 1 / t ? ? (1 + t ?) ? dt =

     .

1/8 ? ((1 + 2t ? + t ?) / t ?) dt =

    .

1/8 ? (1 / t ? + 2 t ? / t ? + t ? / t ?) = 1/8 ? (1 / t ? + 2 / t + t) =

  .

= 1/8 (- 1/2 t ? + 2 ln |t| + t ?/ 2) + C =

   .

1/8 (- 1/2 tg ? x + 2 ln |tg x| + tg ? x / 2) + C =

: - 1/16 tg ? x + 1/4 ln |tg x| + 1/16 tg ? x  + C =

 20.

? dx / (1 + sin x) =

     (1 - sin x).

= ? (1 - sin x) dx / (1 - sin x) (1 + sin x) =

       .

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

          cos ? .

? (1 - sin x) dx / cos ? x =

    .

? (1 / cos ? x - sin x / cos ? x dx =

    .

= ? dx / cos ? x - ? sin x dx / cos ? x =

   ,   .

? dx / cos ? x - ? d(- cos x) / cos ? x = ? dx / cos ? x - ?d cos x / cos ? x =




  .


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   ,     (https://www.litres.ru/book/vasilisa-volkonskaya/integrirovanie-trigonometricheskikh-funktsii-74451408/)  .

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