Интегральное исчисление функции одной переменной. - 12 стр.

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12 §1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .
ôÏÇÄÁ ÉÎÔÅÇÒÁÌ ÏÔ ÄÒÏÂÉ III ÚÁÐÉÛÅÔÓÑ × ×ÉÄÅ:
Z
Ax + B
x
2
+ px + q
dx =
Z
Ax + B
x +
p
2
2
+ k
dx =
Z
A
x +
p
2
−
p
2
+ B
x +
p
2
2
+ k
dx =
=
Z
A
x +
p
2
x +
p
2
2
+ k
dx +
Z
B − A ·
p
2
x +
p
2
2
+ k
dx =
Z
At
t
2
+ k
dt +
Z
B − A ·
p
2
t
2
+ k
dt,
ÇÄÅ t = x +
p
2
.
ðÅÒ×ÙÊ ÉÎÔÅÇÒÁÌ ÌÅÇËÏ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏÄÓÔÁÎÏ×ËÏÊ u = t
2
+ k, Á ÉÍÅÎÎÏ:
Z
t dt
t
2
+ k
=
1
2
Z
dt
2
t
2
+ k
=
1
2
Z
d(t
2
+ k)
t
2
+ k
=
1
2
ln |t
2
+ k| + C.
÷ÔÏÒÏÊ ÉÎÔÅÇÒÁÌ
R
dt
t
2
+k
Ñ×ÌÑÅÔÓÑ ÔÁÂÌÉÞÎÙÍ.
ðÒÉ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÉ ÄÒÏÂÉ IV ÁÎÁÌÏÇÉÞÎÏ III ÐÏÌÕÞÉÍ:
Z
Ax + B
(x
2
+ px + q)
m
dx =
Z
Ax + B dx
x +
p
2
2
+ k
m
=
Z
A
x +
p
2
−
p
2
+ B
x +
p
2
2
+ k
m
dx =
= A
Z
t dt
(t
2
+ k)
m
+
B − A ·
p
2
·
Z
dt
(t
2
+ k)
m
.
ðÅÒ×ÙÊ ÉÎÔÅÇÒÁÌ ÌÅÇËÏ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏÄÓÔÁÎÏ×ËÏÊ u = t
2
+ k, Á ÉÍÅÎÎÏ:
Z
t dt
(t
2
+ k)
m
=
1
2
Z
dt
2
(t
2
+ k)
m
=
1
2
Z
d(t
2
+ k)
(t
2
+ k)
m
=
1
2
Z
du
u
m
=
= −
1
2
·
1
m − 1
·
1
u
m−1
+ C = −
1
2(m − 1)
·
1
(t
2
+ k)
m−1
+ C.
÷ÔÏÒÏÊ ÉÎÔÅÇÒÁÌ ×ÙÞÉÓÌÑÅÔÓÑ ÍÅÔÏÄÏÍ ÐÏÎÉÖÅÎÉÑ. ðÕÓÔØ I
m
=
R
dt
(t
2
+k)
m
,
ÔÏÇÄÁ I
m−1
=
R
dt
(t
2
+k)
m−1
.
òÁÓÓÍÏÔÒÉÍ ÉÎÔÅÇÒÁÌ I
m
:
I
m
=
Z
dt
(t
2
+ k)
m
=
1
k
Z
k
(t
2
+ k)
m
dt =
1
k
Z
k + t
2
− t
2
(t
2
+ k)
m
dt =
=
1
k
·
Z
k + t
2
(t
2
+ k)
m
dt −
Z
t
2
(t
2
+ k)
m
dt
=
=
1
k
Z
dt
(t
2
+ k)
m−1
−
Z
t ·
t dt
(t
2
+ k)
m
=
1
k
·
I
m−1
−
1
2
Z
t
d(t
2
+ k)
(t
2
+ k)
m
.
12                                                §1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .

ôÏÇÄÁ ÉÎÔÅÇÒÁÌ ÏÔ ÄÒÏÂÉ III ÚÁÐÉÛÅÔÓÑ × ×ÉÄÅ:

                                                A x + p2 − p2 + B
                                                             
         Ax + B               Ax + B
  Z                      Z                    Z
                    dx =           2    dx =            2       dx =
       x2 + px + q           x + p2 + k            x + p2 + k
           A x + 2p              B − A · p2                      B − A · p2
                     
                                                     At
       Z                     Z                   Z             Z
     =           2     dx +          2    dx =          dt +              dt,
           x + p2 + k           x + p2 + k         t2 + k          t2 + k

ÇÄÅ t = x + 2p .
   ðÅÒ×ÙÊ ÉÎÔÅÇÒÁÌ ÌÅÇËÏ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏÄÓÔÁÎÏ×ËÏÊ u = t2 + k, Á ÉÍÅÎÎÏ:
                t dt    1      dt2     1 d(t2 + k) 1
          Z               Z               Z
               2
                     =        2
                                     =           2
                                                       = ln |t2 + k| + C.
             t +k       2 t +k         2        t +k       2
÷ÔÏÒÏÊ ÉÎÔÅÇÒÁÌ t2dt+k Ñ×ÌÑÅÔÓÑ ÔÁÂÌÉÞÎÙÍ.
                     R

   ðÒÉ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÉ ÄÒÏÂÉ IV ÁÎÁÌÏÇÉÞÎÏ III ÐÏÌÕÞÉÍ:

                                                          A x + 2p − p2 + B
                                                                          
         Ax + B                    Ax + B dx
  Z                         Z                         Z
                       dx =                      m =                       m dx =
     (x2 + px + q)m
                                                         
                                          2                           2
                                   x + p2 + k                 x + p2 + k
                                                                    

                                               t dt                 p           dt
                                       Z                                 Z
                                  =A                 +    B  − A ·      ·              .
                                           (t2 + k)m                2        (t2 + k)m
ðÅÒ×ÙÊ ÉÎÔÅÇÒÁÌ ÌÅÇËÏ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏÄÓÔÁÎÏ×ËÏÊ u = t2 + k, Á ÉÍÅÎÎÏ:

        t dt       1          dt2         1      d(t2 + k)       1    du
  Z                  Z                      Z                      Z
                 =                     =                      =           =
    (t2 + k)m      2 (t2 + k)m            2 (t2 + k)m            2 um
                           1       1         1                     1           1
                     =− ·               · m−1 + C = −                    · 2          + C.
                           2 m−1 u                            2(m − 1) (t + k)m−1
                                                                                    R dt
÷ÔÏÒÏÊ ÉÎÔÅÇÒÁÌ ×ÙÞÉÓÌÑÅÔÓÑ ÍÅÔÏÄÏÍ ÐÏÎÉÖÅÎÉÑ. ðÕÓÔØ I m = (t2 +k)                         m,
                     dt
                R
ÔÏÇÄÁ Im−1 = (t2 +k)    m−1 .

   òÁÓÓÍÏÔÒÉÍ ÉÎÔÅÇÒÁÌ Im:

               dt         1           k             1      k + t2 − t 2
       Z                     Z                         Z
 Im =                 =                      dt =                       dt =
           (t2 + k)m     k      (t2 + k)m           k       (t2 + k)m
                                    k + t2                    t2
                           Z                                          
                       1
                                                     Z
                    = ·                        dt −                  dt =
                       k          (t2 + k)m              (t2 + k)m
                                                                               d(t2 + k)
          Z                                                                           
      1             dt                      t dt           1              1
                                 Z                                          Z
    =                         − t· 2                    = · Im−1 −            t 2          .
      k        (t2 + k)m−1              (t + k)m           k              2    (t + k)m

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