Неопределенные интегралы. Желтухин В.С. - 68 стр.

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sh x ch y =
1
2
[sh(x − y) + sh(x + y)];
sh(x + y) sh(x − y) = ch
2
x − ch
2
y = sh
2
x − sh
2
y;
ch(x + y) ch(x − y) = sh
2
x + ch
2
y = ch
2
x + sh
2
y.
(ch x ± sh y)
n
= ch nx ± sh nx.
sh
2
x =
1
2
(ch 2x − 1); ch
2
x =
1
2
(ch 2x + 1);
y = Arsh x , x = sh y;
y = Arch x , x = ch y;
y = Arth x , x = th y;
y = Arcth x , x = cth y.
y = ch x
y = Arch x
y = Arsh x = ln
³
x +
p
x
2
+ 1
´
;
y = Arch x =
ln
³
x −
p
x
2
− 1
´
, x ≥ 1 − ∞ < y ≤ 0;
ln
³
x +
p
x
2
− 1
´
, x ≥ 1 0 ≤ y ≤ +∞;
y = Arth x = ln
r
1 + x
1 − x
=
1
2
ln
1 + x
1 − x
|x| < 1;
                              1
                                [sh(x − y) + sh(x + y)];
                      sh x ch y =
                              2
        sh(x + y) sh(x − y) = ch2 x − ch2 y = sh2 x − sh2 y;
        ch(x + y) ch(x − y) = sh2 x + ch2 y = ch2 x + sh2 y.

Ôîðìóëà Ìóàâðà


                      (ch x ± sh y)n = ch nx ± sh nx.

Ôîðìóëû ïîíèæåíèÿ ñòåïåíè


                      1                                1
            sh2 x =     (ch 2x − 1);         ch2 x =     (ch 2x + 1);
                      2                                2
Îáðàòíûå ãèïåðáîëè÷åñêèå ôóíêöèè



       y=                    Arsh x (àðåàñèíóñ),          åñëè x = sh y;
       y=                Arch x (àðåàêîñèíóñ),            åñëè x = ch y;
       y=                Arth x (àðåàòàíãåíñ),            åñëè x = th y;
       y=         Arcth x (àðåàêîòàíãåíñ),               åñëè x = cth y.

Ñëåäóåò ó÷åñòü, ÷òî y = ch x íå âî âñåé îáëàñòè îïðåäåëåíèÿ  ìî-
íîòîííàÿ ôóíêöèÿ. Ïîýòîìó äëÿ êàæäîãî èç äâóõ èíòåðâàëîâ ìîíî-
òîííîñòè ïîëó÷àþò ñâîþ îáðàòíóþ ôóíêöèþ y = Arch x.

Ñâÿçü îáðàòíûõ ãèïåðáîëè÷åñêèõ ôóíêöèé ñ ëîãàðèôìè÷åñêîé ôóíê-
öèåé


                 ³       p          ´
 y = Arsh x = ln x +    +1 ;x2
               ³     p     ´
              ln x − x2 − 1 , äëÿ x ≥ 1 è − ∞ < y ≤ 0;
 y = Arch x =     ³   p     ´
              ln x + x2 − 1 , äëÿ x ≥ 1 è 0 ≤ y ≤ +∞;
                 r
                   1+x 1 1+x
 y = Arth x = ln      = ln       ïðè |x| < 1;
                   1−x 2 1−x
                                        68

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