Неопределенные интегралы. Желтухин В.С. - 69 стр.

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y = Arcth x = ln
r
x + 1
x − 1
=
1
2
ln
x + 1
x − 1
|x| > 1.
a = sh x a = ch x a = th x a = cth x
sh x ±
p
a
2
− 1 ±
a
p
1 − a
2
±
1
p
a
2
− 1
ch x
p
a
2
+ 1
1
p
1 − a
2
a
p
a
2
− 1
th x ±
a
p
a
2
+ 1
±
p
a
2
− 1
a
1
a
cth x ±
p
a
2
+ 1
a
±
a
p
a
2
− 1
1
a
sh x = a cth x =
p
a
2
+ 1
a
(x ≥ 0) Arsh a = Arcth
p
a
2
+ 1
a
Arsh x ± Arch y = Arsh
³
xy ±
p
(1 + x
2
)(y
2
− 1)
´
=
= Arch
h
y
p
1 + x
2
± x
p
y
2
− 1
i
;
Arsh x ± Arsh y = Arsh
³
x
p
1 + y
2
± y
p
1 + x
2
´
;
Arch x ± Arch y = Arsh
³
xy ±
p
(x
2
− 1)(y
2
− 1)
´
;
Arth x + Arth y = Arth
x ± y
1 ± xy
;
Arcth x ± Arcth y = Arcth
1 ± xy
x ± y
.
                    r
                        x+1 1 x+1
y = Arcth x = ln           = ln         ïðè |x| > 1.
                        x−1 2 x−1

Ñîîòíîøåíèÿ ìåæäó ãèïåðáîëè÷åñêèìè (èëè îáðàòíûìè ãèïåðáîëè-
÷åñêèìè) ôóíêöèÿìè

             a = sh x      a = ch x     a = th x            a = cth x
                           p              a                        1
     sh x       a         ± a2 − 1     ±p                   ±p
                                         1 − a2                 a2 − 1
            p                                1                 a
     ch x    a2 + 1           a
                                        p                   p
                                            1 − a2           a2 − 1
                            p
                    a        a2 − 1                            1
     th x   ±p            ±                  a
                              a                                a
                 a2 + 1
                p
                 a2 + 1      a               1
    cth x   ±             ±p                                   a
                  a                          a
                            a2 − 1
                          p                               p
                           a2 + 1                          a2 + 1
Åñëè sh x = a, òî cth x =         (x ≥ 0), Arsh a = Arcth         .
                            a                               a
Çíàê âûáèðàåòñÿ â ñîîòâåòñòâèè ñî çíàêîì ëåâîé ÷àñòè.

Ñóììà è ðàçíîñòü îáðàòíûõ ãèïåðáîëè÷åñêèõ ôóíêöèé

                                  ³    p                         ´
        Arsh x ± Arch y = Arsh xy ± (1 +         x2 )(y 2
                                                − 1) =
                              h p         p         i
                                      2      2
                        = Arch y 1 + x ± x y − 1 ;
                              ³ p         p         ´
                                      2
        Arsh x ± Arsh y = Arsh x 1 + y ± y 1 + x ;2
                              ³    p                  ´
                                        2     2
        Arch x ± Arch y = Arsh xy ± (x − 1)(y − 1) ;
                                x±y
        Arth x + Arth y = Arth        ;
                               1 ± xy
                                1 ± xy
      Arcth x ± Arcth y = Arcth         .
                                 x±y

                                  69

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