Неопределенные интегралы. Желтухин В.С. - 41 стр.

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p = b/a x = t − p/2
J
k
=
A
k
α
Z
αt dt
(t
2
+ λ)
k
p
αt
2
+ β
+ B
k
Z
dt
(t
2
+ λ)
k
p
αt
2
+ β
.
u =
p
αt
2
+ β
J =
Z
(x + 3) dx
(x
2
− x + 1)
p
x
2
+ x + 1
x =
µt + ν
t + 1
,
x
2
± x + 1 =
(µ
2
± µ + 1)t
2
+ [2µν ± (µ + ν) + 2]t + (ν
2
± ν + 1)
(t + 1)
2
.
2µν ± (µ + ν) + 2 = 0,
µ + ν = 0, µν = −1,
µ = 1 ν = −1
x =
t − 1
t + 1
, dx =
2 dt
(t + 1)
2
, x + 3 =
4t + 2
t + 1
,
x
2
− x + 1 =
t
2
+ 3
(t + 1)
2
,
p
x
2
+ x + 1 =
p
3t
2
+ 1
t + 1
,
t + 1 > 0 x < 1
J =
Z
(8t + 4) dt
(t
2
+ 3)
p
3t
2
+ 1
=
= 8
Z
t dt
(t
2
+ 3)
p
3t
2
+ 1
+ 4
Z
dt
(t
2
+ 3)
p
3t
2
+ 1
.
     2b) Åñëè p = b/a, òî ëèíåéíàÿ çàìåíà x = t − p/2 ñðàçó ïðè-
âîäèò èíòåãðàë (23) ê èíòåãðàëó âèäà (27).

       Ïîëó÷åííûé èíòåãðàë ðàçëàãàåòñÿ íà äâà:
                 Z                                Z
            Ak              αt dt                 dt
       Jk =                   p        + Bk        p         .
            α        (t2 + λ)k αt2 + β   (t2 + λ)k αt2 + β
                                               p
Ïåðâûé èç íèõ ëåãêî áåðåòñÿ ïîäñòàíîâêîé u = αt2 + β . Êî âòî-
ðîìó ïðèìåíÿþòñÿ ïîäñòàíîâêè (21) èëè (22).
                                Z
                                          (x + 3) dx
Ï ð è ì å ð 46. Âû÷èñëèòü J =                  p           .
                                   (x2 − x + 1) x2 + x + 1
     . Äðîáíî-ëèíåéíàÿ ïîäñòàíîâêà
                                       µt + ν
                                  x=          ,
                                        t+1
äàåò

  2      (µ2 ± µ + 1)t2 + [2µν ± (µ + ν) + 2]t + (ν 2 ± ν + 1)
 x ±x+1=                                                       .
                               (t + 1)2
       Òðåáîâàíèÿ
                           2µν ± (µ + ν) + 2 = 0,
èëè
                           µ + ν = 0,       µν = −1,
óäîâëåòâîðÿþòñÿ, íàïðèìåð, ïðè µ = 1, ν = −1. Èìååì
                 t−1              2 dt              4t + 2
            x=       ,     dx =         ,   x + 3 =        ,
                 t+1           (t + 1)2             t+1
                                                     p
                          t2
                             + 3       p                3t2 + 1
            x2 − x + 1 =          ,      x 2+x+1 =              ,
                         (t + 1)2                       t+1
åñëè, äëÿ îïðåäåëåííîñòè, ñ÷èòàòü t + 1 > 0 (ò.å. x < 1). Òàêèì
îáðàçîì,
        Z
                 (8t + 4) dt
 J=                  p          =
             (t2 + 3) 3t2 + 1
                             Z                     Z
                                      t dt                   dt
                        =8             p        +4           p        .
                               (t2 + 3) 3t2 + 1      (t2 + 3) 3t2 + 1

                                       41

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