  
 


              .      ,         .       .       ,    ,     .





 

  



:

1)  .

2)       .

3)    .

4) ׸     .

5)   .

6)      .

7)    .    .

8)   .

9)      .

10)     .

11)   .

12)   .

13)        .

14)     .

15)   .  .

16)      .

17)   .

18)    .

18. 1       .

18. 2        .

19)       .

20)    ,      .



  .

1))  .

  -    ,         .

 -      .

 -  .

 ? 2, 718

f (x) = e * - .

   .

 .

sh x = ( *-  ?*) / 2 = (e ?* - 1) / 2e*

 .

ch x = ( *+  ?*) / 2 = (e ?* + 1) / 2e*

 .

th x = sh x / ch x = (e * -  ?*) / (e * +  ?*) = (e?* - 1) / (e?* + 1)

 .

cth x = 1 / th x = ch x / sh x = (e * +  ?*) / (e * -  ?*) = (e ?* + 1) / (e ?* - 1)

 .

sch x = 1 / ch x = 2 / (e * +  ?*)

 .

csch x = 1 / sh x = 2 / (e * -  ?*)

  -      .

  -      .

2))       .

       .

        ,      :

x ? + y ? = R ?

 ? + y ? = 1 ,  R = 1

  x    cos ? ().

      sin ? ().

    .

cos ? ? + sin ? ? = 1

      .








           ,      x;     ,      x.

         .

      .

       ,       .

x ? / a ? - y ? / b ? = 1



 -  .

b -  .

  a  b  ,     .

x ? - y ? = 1

 x   .

  y   .

  x      ch t.

  y      sh t.

   .

ch ? t - sh ? t = 1

t -   ,     ,    ,      x;     ,      x.















t -   ,   .

      .

   ,      .

  -   ,    .

  -   ,    .








     ,   .

        .

           .

3))    .   .

h ? x - sh ? x = 1

     .

(e *+  ?*) ? / 2 - ( * -  ?*) ? / 2 = 1

    .

(( * +  ?*) ? - ( * - e ?*) ?) / 4 = 1

       .

(e ?* + 2 e * ?  ?* +  ??*- ( ?* - 2  *?  ?*+  ??*)) / 4 = 1

(e ?* + 2e *?* +  ??* -  ?* + 2e *?* - e ??*) / 4 = 1

(2e ? + 2e ?) / 4 = 1

4 / 4 = 1

1 = 1

 .

4)) ׸     .

   .

ch (- x) = ch x

 .

 , ,   .

sh (- x) = - sh x

th (- x) = - th x

cth (- x) = - cth x

   .

5))   .

1)  .

y = sh x








       .

D (f): x ? (- ?; + ?).

E(f): y ? (- ?; + ?)

2)  .

y = ch x








    (- ?; 0)    (0; + ?).

 (0; 1)   .

3)  .

y = th x








      .

     y = - 1  y = 1.

D(f): x ? (- ?; +?).

E(f): y ? (- 1;  1)

4)  .

y = cth x








    (- ?; 0); (0; + ?)

     y = 1  y = - 1    x = 0.

D(f): x ? x ? (- ?; 0); (0; +?).

E(f): y ? (- ?; - 1); (1; + ?)

6))      .

        .

sh x = - i sin i x

ch x = cos i x

th x = - i tg i x

cth x = - 1 / i ctg i x = - i ctg x

sh i x = i sin x

ch i x = cos x

th i x = i tg x

cth i x = 1 / i ctg x = - i ctg i x

7))    .

sh x - ch x = -  ?*

ch x + sh x = e *

ch x - sh x = e ?*

ch ? x - sh ? x = 1

ch ? x = 1 + sh ? x

sh ? x = ch ? x - 1

  .

th x = sh x / ch x

cth x = ch x / sh x

1 / ch ? x = 1 - th ? x

1 / sh ? x = cth ? x - 1

th ? x = 1 - 1 / h ? x

cth ? x = 1 + 1 / sh ? x

th x ? cth x = 1

th x = 1 / cth x

cth x = 1 / th x

  .

sh 2x = 2 ch x sh x

ch 2x = ch ? x + sh ? x

ch 2x = 2ch ? x - 1

ch 2x = 1 + 2 sh ? x

th 2x = 2 th x / (1 + th ? x) cth 2x = (1 + th ? x) / 2 th x

cth 2x = (th x + cth x) / 2 = th 2x

       .

sh 2x = 2 th x / (1 - th ? x)

ch 2x = (1 + th ? x) / (1 - th ? x)

  .

ch ? x = 1/2 (ch 2x + 1)

sh ? x  = 1/2 (ch 2x - 1)

th ? x = (ch 2x - 1) /(ch 2x + 1)

cth ? x = (ch 2x + 1) /(ch 2x - 1)

     .

ch ? x = 1 / (1 - th ? x)

sh ? x = 1 / (cth ? x - 1)

ch ? x = cth ? x / (cth ? x - 1)

sh ? x = th ? x / (1 - th ? x)

    .

sh x = 2 (th ( / 2)) / (1 - th ? ( / 2))

ch x = (1 + th ? ( / 2)) / (1 - th ? ( / 2))

th x = (2 th ( / 2)) / (1 + th ? ( / 2))

cth x = (1 + th ? ( / 2))) / 2 th ( / 2))

       .

sh x + sh y = 2 sh (x + y) / 2 ch (x - y) / 2

sh x - sh y = 2 sh (x - y) / 2 h (x + y) / 2

ch x + ch y = 2 ch (x + y) / 2 h (x - y) / 2

ch x - ch y = 2 sh (x + y) / 2 sh (x - y) / 2

.

    ,   .

    ,           ,   .

  ,      .

       .

     ,     .

         .

th x + th y = sh (x + y) /  ch x ch y

th x - th y = sh (x - y) / ch x ch y

cth x + cth y = sh (x + y) / sh x sh y

cth x - cth y = sh (y - x) / sh x sh y

th x + cth y = ch (x + y) /ch x shy

th x - cth y = - ch (x - y) / sh x ch y

cth x - th y = ch (x - y) / sh x ch y

     .

sh (x + y) = sh x ch y + ch x sh y

sh (x - y) = shx ch y - ch x sh y

ch (x + y) = ch x ch y + sh x sh y

ch (x - y) = ch x ch y - sh x sh y

.

       ,   ;    .

       ,        ,     .

        .

th (x + y) = (th x + th y) / (1 + th x th y)

th (x - y) = (th x - th y) / (1 - th x th y)

th (x + y) = (cth x cth y + 1) / (cth x + cth y)

th (x - y) = (cth y - cth x) / (cth x cth y - 1)

cth (x + y)= (1 + cth x cth y) / (cth x + cth y)

cth (x - y) = (1 - cth x cth y) / (cth x - cth y)

cth (x + y) = (1 + th x th y) / (th x + th y)

cth (x - y) = (th x th y - 1) / (th y - th x)

  .

sh x sh y = 1/2 ((ch (x + y) - ch (x - y))

ch x ch y = 1/2 ((ch (x + y) + ch (x - y))

sh x ch y = 1/2 ((sh (x + y) + sh (x - y))

th x th y = (ch (x + y) - ch (x - y)) /(ch (x + y) + h ( - y))

cth x cth y = (ch (x + y) + ch (x - y)) / (ch (x + y) - ch (x - y))

         .

th x = (ch 2x - 1) / sh 2x

th x = sh 2x / (1 + ch 2x)

cth x = sh 2x / (ch 2x - 1)

cth x = (1 + ch 2x) / sh 2x

ch 2x + sh 2x =  (sh x + ch x) ?

ch 2x - sh 2x = (sh x - ch x) ?

  .

sh 3x = 4 sh ? x + 3 sh x

ch 3x = 4 ch ? x - 3 ch x

th 3X = th x (3 + th ? x) / (1 + 3 th ? x)

cth 3x = (1 + 3 th ? x) / th x (3 + th ? x)

sh 5x = 16 sh ? x + 20 sh ? x + 5 shx

ch 5x = 16 ch ? x - 20 ch ? x + 5 ch x

th 5x = th x (th ? x + 10 th ? x + 5) / (5 th ? x + 10 th ? x + 1)

  .

sh ? ( / 2) = (ch x - 1) / 2

ch ? ( / 2) = (ch x + 1) / 2

th ( / 2) = sh x / (ch x + 1) = (ch x - 1) / sh x

cth ( / 2) = (ch x + 1) /  sh x = sh x / (ch x - 1)

8))   .

arch x -   .

arsh x -  .

arth x -  .

arcth x -  .

    ar  x,   ar  area - ;      arc  x,   arc -  .

  , , ,   , , , ,   .

1) y = arch x - .








D (f): x ? [1; + ?)

E (f): y ? [0; + ?)

 (1; + ?)

2) y = arsh x - .








D (f): x ? (- ?; + ?)

E (f): y ? (- ?; + ?)

  (- ?; + ?)

 .

arsh (- x) = - arsh x



3) y = arth x - .








D (f): x ? (- 1; 1)

E (f): y ? (- ?; + ?)

  (- ?; + ?)

 .

arth (- x) = - arth x

4) y = arcth x - .








D (f): x ? (- ?; - 1); (1; + ?)

E (f): y ? (- ?; 0); (0; + ?)

  (- ?; - 1); (1; + ?)

 .

arcth (- x) = - arcth x

     .

1)   -      .

sch x = 1 / ch x = 2 / ( * +  ?*)








 .

sch (- x) = sch x

 .

D (f): x ? (- ?; + ?)

E (f): y ? (0; + 1]

   (- ?; 0)    (0; + ?)

2)   -      .

csch x = 1 / sh x = 2 / ( * -  ?*)








 .

csch (- x) = - csch x

   .

D (f): x ? (- ?; 0); (0; + ?)

E (f): y ? (- ?; 0); (0; + ?)

9))      .

sh (arsh a) = a

ch (arch a) = a

th (arth a) = a

cth (atcth a) = a

arch (sh x) = x

arch (ch x) = x

arth (th x) = x

arcth (cth x) = x

arch (- a) = arch a

arsh (- a) = - arsh a

arth (- a) = - arth a

arcth (- a) = - arcth a

sh (arsh x) = ?(x ? - 1)   |x| > 1

ch (arsh x) = ?(x ? + 1);    ? R

sh (arth x) = x / ?(1 - x ?);   |x| < 1

ch (arth x) = 1 / ?(1 - x ?);    0 ? x ? 1

sh (arcth x) = 1 / ?(x ? - 1);   |x| > 1

ch (arcth x) = x / ?(x ? - 1);   x > 1

th (arch x) =  ?(x ? - 1) / x;   x ? 1

th (arsh x) = x / ?(x ? + 1);   x ? R

cth (arch x) = x / ?(x ? - 1);   x > 1

cth (arsh x) = ?(x ? + 1) / x;        x ? 0

10))     .

1) arsh x =  ln |x + ?(x ? + 1);  x ? R

2) arsh x = |x + ?(x ? - 1)|;   x ? 1

3) arth x = 1/2 ln |(1 + x) / (1 - x)|;          |x| < 1

4) arcth x = 1/2 ln |(x + 1) / (x - 1);          |x| > 1

5) ln x = arth ((x ? - 1) / (x ? + 1)) = sh ((x ? - 1) / 2x) =  arch ((x ? + 1) / 2x)

6) arth x = arsh (x / ?(1 - x ?)) =  arch (1 / ?(1 - x ?))

7) arsh x = arth (x / ?(1 + x ?)) = arsh  ?(1 + x ?)

8) arch x = |arsh ?(x ? - 1)| = |arth ?(x ? - 1) / x|

11))   .

(sh x) ' = ch x

(ch x) ' = sh x

(th x) ' = 1 /  ch ? X

(cth x) ' = - 1 / sn ? x

(arsh x) ' = 1 / ?(x ? + 1)

(arch x) ' = 1 / ?(x ? - 1)

(rth x) ' = 1 / (1 - x ?)

(arcth x) ' = 1 / (x ? - 1)

(ln|ch x|) ' = thx

(ln |sh x|) ' = cth x

(arsh (x / a)) ' = 1 / ?(x ? + a ?)

(rch (x / a)) ' = 1 / ?(x ? - a ?)

(rth (x / a)) ' = a / (a ? - x ?)

(arcth (x / a)) ' = a / (a ? - x ?)

(rcsch (x / a)) ' = - a / x ?( ? - x ?)

(arsh (x / a)) ' = - a / x ?(a ? + x ?)

(csh x) ' = - cthx csch x

(csh x) ' = - th x sch x

12))   .

? ch x dx = sh x + C

? sh x dx = ch x + C

? dx / ch ? x = th x + C

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

? dx /?(x ? + 1) = arsh x + C

? dx / ?(x ? - 1) = arch x + C

? dx / (1 - x ?) = arth x + C

? dx / (x ? - 1) = - arcth x + C

? th x dx = ln |ch x| + C

? cth x dx = ln |sh x| + C

? dx /?(x ? + a ?) = arsh (x / a) + C

? dx / ?(x ? - a ?) = arch (x / a) + C

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

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

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

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

? cth x csch x dx = - csch x + C

? th x sch x dx = - sch x + C

13))        .

? dx / ?(x ? + 1) = arsh x + C = ln |x + ?(x? + 1)| + C; x ? R

? dx / ?(x ? - 1) = arch x + C = ln |x + ?(x ? - 1)| + C; x ? 1

? dx / (1 - x ?) = arth x + C = 1 / 2 ln |(1 + x) / (1 - x)| + C; |x| < 1

? dx /(x ? - 1) = - arcth x + C = 1/2 ln |(x - 1) / (x + 1)| + C; |x| > 1

? dx / ?(x ? + a ?) = arsh (x / a) + C = ln |x + ?(x ? + a ?)| + C; x / a > 1;  ? 0

? dx / ?( ? - a ?) = arch (x / a) + C = ln |(x + ?(x ? - a ?)| + C; x / a > 1; a ? 0

? dx / (a ? - x ?) = 1 / a arth (x / a) + C = 1 / (2) ln |( + x) / (a - x)| + C; x / a < 1;  ? 0

? dx / (x ? - a ?) = 1 / a arcth (x / a) + C = 1 / (2) ln |( - x) / (a + x)| + C; x / a > 1;  ? 0

? csch x dx = ln |th (x / 2)| + C = ln |(e * - 1) / (e * + 1)| + C

? sch x dx = 2 arctg th (x / 2) + C = 2 arctg  * + C

14))     .

   .

 .

? dx / ch x =

      .

= ? ch x dx / ch ? x =

     .   .

= ? d sh x / ch ? x =

         .

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

  .

 sh x = t

= ? dt / (1 + t ?) =

  .

    .

= arctg t + C =

  .

= arctg sh x + C

: arctg sh x + C

 .

? dx / sh x =

       .

= ? sh x dx / sh ? x =

      .

   .

= ? d ch x / sh ? x =

           .

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

   .

 ch x = t

= ? dt / (t ? - 1) =

   .

     .

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




  .


   .

   ,     (https://www.litres.ru/book/vasilisa-volkonskaya/integrirovanie-giperbolicheskikh-funktsii-74505553/)  .

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