Интегральное исчисление функции одной переменной. - 6 стр.

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6 §1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .
ðÒÉÍÅÒ 11.
Z
e
2x
e
2x
+ 1
dx =
1
2
Z
e
2x
e
2x
+ 1
d2x =
1
2
Z
de
2x
e
2x
+ 1
=
1
2
Z
d(e
2x
+ 1)
e
2x
+ 1
=
=
1
2
Z
dt
t
=
1
2
ln |t| + C =
1
2
ln(e
2x
+ 1) + C.
ðÒÉÍÅÒ 12.
Z
x dx
5x
2
− 3
=
1
2
Z
dx
2
5x
2
− 3
=
1
2 · 5
Z
d(5x
2
− 3)
5x
2
− 3
=
=
1
10
Z
dt
t
=
1
10
ln |t| + C =
1
10
ln |5x
2
− 3| + C.
II. ðÒÉÍÅÎÑÑ ×ÔÏÒÕÀ ÆÏÒÍÕ ÐÏÄÓÔÁÎÏ×ËÉ, ÐÏÌØÚÕÀÔÓÑ ÐÒÁ×ÉÌÏÍ 4. ÷ ÐÏ-
ÄÙÎÔÅÇÒÁÌØÎÏÅ ×ÙÒÁÖÅÎÉÅ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏ ÐÏÄÓÔÁ×ÌÑÀÔ ×ÍÅÓÔÏ x ÆÕÎËÃÉÀ
x = ω(t), Á ÉÍÅÎÎÏ:
Z
f(x) dx =
Z
f(ω(t)) dω(t) =
Z
f(ω(t))ω
0
(t) dt =
Z
g(t) dt,
ÇÄÅ g(t) ¡ ÂÏÌÅÅ ÕÄÏÂÎÁÑ ÄÌÑ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÑ ÆÕÎËÃÉÑ, ÞÅÍ f (x). ðÒÉ ÜÔÏÍ
ÎÁ ÆÕÎËÃÉÀ ω(t) ÎÁËÌÁÄÙ×ÁÀÔÓÑ ÕÓÌÏ×ÉÑ ÓÔÒÏÇÏÊ ÍÏÎÏÔÏÎÎÏÓÔÉ, ÞÔÏ ÏÂÅÓ-
ÐÅÞÉ×ÁÅÔ ÓÕÝÅÓÔ×Ï×ÁÎÉÅ ÏÂÒÁÔÎÏÊ ÆÕÎËÃÉÉ t = v(x) É ÐÒÅÄÓÔÁ×ÌÅÎÉÅ:
Z
f(x) dx =
Z
g(t) dt = G(t) + C = G(v(x)) + C.
ðÒÉÍÅÒ 13.
Z
dx
√
x (4 +
3
√
x)
.
ðÒÉÍÅÎÑÑ ÐÏÄÓÔÁÎÏ×ËÕ x = t
6
, ÐÏÌÕÞÉÍ
√
x = t
3
,
3
√
x = t
2
, dx = dt
6
= 6t
5
dt
É
Z
6t
5
dt
t
3
(4 + t
2
)
= 6
Z
t
2
4 + t
2
dt = 6
Z
t
2
+ 4 − 4
t
2
+ 4
dt =
= 6
Z
1 −
4
t
2
+ 4
dt = 6
Z
dt − 24
Z
dt
t
2
+ 4
=
= 6t − 12 arctg
t
2
+ C = 6
6
√
x + 12 arctg
6
√
x
2
+ C.
ðÒÉÍÅÒ 14.
Z
p
9 − x
2
dx.
6                                         §1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .

     ðÒÉÍÅÒ 11.
        e2x        1   e2x         1     de2x     1 d(e2x + 1)
    Z                Z               Z             Z
              dx =           d2x =            =                =
      e2x + 1      2 e2x + 1       2 e2x + 1 2         e2x + 1
                                 1 dt 1                 1
                                   Z
                               =         = ln |t| + C = ln(e2x + 1) + C.
                                 2     t   2            2
     ðÒÉÍÅÒ 12.
       x dx    1   dx2    1       d(5x2 − 3)
    Z            Z            Z
             =         =                       =
      5x2 − 3 2 5x2 − 3 2 · 5      5x2 − 3
                         1     dt     1               1
                            Z
                       =           =     ln |t| + C =    ln |5x2 − 3| + C.
                         10     t    10               10
   II. ðÒÉÍÅÎÑÑ ×ÔÏÒÕÀ ÆÏÒÍÕ ÐÏÄÓÔÁÎÏ×ËÉ, ÐÏÌØÚÕÀÔÓÑ ÐÒÁ×ÉÌÏÍ 4. ÷ ÐÏ-
ÄÙÎÔÅÇÒÁÌØÎÏÅ ×ÙÒÁÖÅÎÉÅ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏ ÐÏÄÓÔÁ×ÌÑÀÔ ×ÍÅÓÔÏ x ÆÕÎËÃÉÀ
x = ω(t), Á ÉÍÅÎÎÏ:
        Z            Z                Z                  Z
                                                0
           f (x) dx = f (ω(t)) dω(t) = f (ω(t))ω (t) dt = g(t) dt,

ÇÄÅ g(t) ¡ ÂÏÌÅÅ ÕÄÏÂÎÁÑ ÄÌÑ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÑ ÆÕÎËÃÉÑ, ÞÅÍ f (x). ðÒÉ ÜÔÏÍ
ÎÁ ÆÕÎËÃÉÀ ω(t) ÎÁËÌÁÄÙ×ÁÀÔÓÑ ÕÓÌÏ×ÉÑ ÓÔÒÏÇÏÊ ÍÏÎÏÔÏÎÎÏÓÔÉ, ÞÔÏ ÏÂÅÓ-
ÐÅÞÉ×ÁÅÔ ÓÕÝÅÓÔ×Ï×ÁÎÉÅ ÏÂÒÁÔÎÏÊ ÆÕÎËÃÉÉ t = v(x) É ÐÒÅÄÓÔÁ×ÌÅÎÉÅ:
             Z            Z
                f (x) dx = g(t) dt = G(t) + C = G(v(x)) + C.

    ðÒÉÍÅÒ 13.
                                           dx
                               Z
                                     √        √ .
                                       x (4 + 3 x)
                                              √        √
ðÒÉÍÅÎÑÑ ÐÏÄÓÔÁÎÏ×ËÕ x = t6 , ÐÏÌÕÞÉÍ x = t3 , 3 x = t2 , dx = dt6 = 6t5 dt
É
       6t5 dt            t2
                                      Z 2
                                         t +4−4
  Z                  Z
                 = 6          dt =  6              dt =
    t3 (4 + t2 )       4 + t2              t2 + 4
                     Z             
                                4                             dt
                                               Z        Z
                 =6      1− 2          dt = 6 dt − 24              =
                              t +4                          t2 + 4
                                                                     √
                                                 t       √           6
                                                                       x
                               = 6t − 12 arctg + C = 6 x + 12 arctg
                                                          6
                                                                         + C.
                                                2                     2
   ðÒÉÍÅÒ 14.                     Z p
                                        9 − x2 dx.

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