Неопределенные интегралы. Желтухин В.С. - 13 стр.

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Z
¡
αx
2
+ β
¢
µ
x dx, (µ 6= −1)
t = x
2
u = αx
2
+
β x dx du =
2αx dx
Z
¡
αx
2
+ β
¢
µ
x dx =
1
2α
Z
u
µ
du =
1
2α(µ + 1)
u
µ+1
+ C =
=
1
2α(µ + 1)
¡
αx
2
+ β
¢
µ+1
+ C.
Z
g(ln x)
dx
x
=
Z
g(ln x)d ln x
t = ln x
Z
ln x
x
dx =
Z
ln x d ln x =
1
2
ln
2
x + C.
Z
g(sin x) cos x dx,
Z
g(cos x) sin x dx,
Z
g(tg x)
dx
cos
2
x
,
t = sin x, t = cos x, t = tg x.
Z
cos x dx
1 + sin
2
x
=
Z
dt
1 + t
2
= arctg sin x + C.
Z
f
0
(x)
f(x)
dx =
Z
d f(x)
f(x)
t = f(x)
Z
2x dx
x
2
+ 1
=
Z
d(x
2
+ 1)
x
2
+ 1
= ln(x
2
+ 1) + C
Z
dx
sin x cos x
=
Z
1
tg x
dx
cos
2
x
=
Z
d(tg x)
cos
2
x
= ln |tg x| + C. 
                      Z
                          ¡ 2    ¢µ
Ï ð è ì å ð 21. .          αx + β x dx, (µ 6= −1)  â ýòîì èíòåãðàëå
ìîæíî áûëî áû ïîëîæèòü t = x2 , íî ïðîùå ñðàçó âçÿòü u = αx2 +
β , òàê êàê ìíîæèòåëü x dx ëèøü ìíîæèòåëåì îòëè÷àåòñÿ îò du =
2αx dx. Òàêèì îáðàçîì, èìååì
    Z                                      Z
         ¡ 2    ¢µ                     1                   1
          αx + β x dx =                        uµ du =           uµ+1 + C =
                                      2α               2α(µ + 1)
                                          1         ¡ 2     ¢µ+1
                              =                      αx + β      + C. /
                                      2α(µ + 1)
                                                    Z
                                                    Z
                                             dx
Ï ð è ì å ð 22. . Èíòåãðàëû âèäà g(ln x) = g(ln x)d ln x
                                              x
áåðóòñÿ ïîäñòàíîâêîé t = ln x. Íàïðèìåð,
             Z           Z
                ln x                   1
                     dx = ln x d ln x = ln2 x + C. /
                 x                     2
Ï ð è ì å ð 23. . Èíòåãðàëû âèäà
     Z                            Z                            Z
                                                                                dx
         g(sin x) cos x dx,            g(cos x) sin x dx,            g(tg x)          ,
                                                                               cos2 x
áåðóòñÿ, ñîîòâåòñòâåííî, ïîäñòàíîâêàìè

                     t = sin x,           t = cos x,     t = tg x.

Íàïðèìåð,
              Z                       Z
                   cos x dx         dt
                             =           = arctg sin x + C. /
                  1 + sin2 x     1 + t2
                                         Z 0            Z
                                            f (x)           d f (x)
Ï ð è ì å ð 24. . Èíòåãðàëû âèäà                  dx =              (÷èñëè-
                                            f (x)            f (x)
òåëü ïðåäñòàâëÿåò ñîáîé äèôôåðåíöèàë çíàìåíàòåëÿ) ñðàçó áåðóòñÿ
ïîäñòàíîâêîé t = f (x). Íàïðèìåð,
     Z            Z
        2x dx        d(x2 + 1)
  1)     2
               =        2
                                = ln(x2 + 1) + C .
       x +1           x +1
     Z                Z                  Z
           dx              1    dx         d(tg x)
  2)                =             2
                                       =            = ln | tg x| + C. /
       sin x cos x        tg x cos x        cos2 x



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