Интегральное исчисление функции одной переменной. - 24 стр.

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24 §1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .
Ç) m < 0, |m| = 2k.
Z
cos
m
x dx =
Z
dx
cos
2k
x
=
Z
1
cos
2
x
k−1
dx
cos
2
x
=
=
Z
sin
2
x + cos
2
x
cos
2
x
k−1
d tg x =
Z
(1 + tg
2
x)
k−1
d tg x =
=
Z
(1 + t
2
)
k−1
dt, t = tg x,
ÁÎÁÌÏÇÉÞÎÏ
Z
sin
m
x dx =
Z
dx
sin
2k
x
= −
Z
(1 + ctg
2
x)
k−1
d ctg x.
ðÒÉÍÅÒ 37.
Z
dx
cos
4
x
=
Z
1
cos
2
x
dx
cos
2
x
=
Z
sin
2
x + cos
2
x
cos
2
x
d tg x =
Z
(tg
2
x + 1) d tg x =
=
Z
(t
2
+ 1) dt =
t
3
3
+ t + C =
tg
3
x
3
+ tg x + C.
Ä) m < 0, |m| = 2k + 1.
Z
cos
m
x dx =
Z
dx
cos
2k+1
x
=
Z
cos x
cos
2k+2
x
dx =
=
Z
d sin x
(1 − sin
2
x)
k+1
=
Z
dt
(1 − t
2
)
k+1
ÄÁÌÅÅ ÓÍ. ÐÕÎËÔ 1.4. óÌÕÞÁÊ
R
dx
sin
2k+1
x
ÄÅÌÁÅÔÓÑ ÁÎÁÌÏÇÉÞÎÏ.
ðÒÉÍÅÒ 38.
Z
dx
cos
3
x
=
Z
cos x dx
cos
4
x
=
Z
d sin x
(cos
2
x)
2
=
Z
d sin x
(1 − sin
2
x)
2
=
Z
dt
(1 − t
2
)
2
;
(ÄÁÌÅÅ ÓÍ. ÐÕÎËÔ 1.4).
÷ ÓÌÕÞÁÅ Ä) ÄÌÑ ÐÏÎÉÖÅÎÉÑ ÓÔÅÐÅÎÉ ÕÄÏÂÎÏ ÐÒÉÍÅÎÑÔØ ÆÏÒÍÕÌÕ sin
2
x +
+ cos
2
x = 1.
Z
dx
cos
3
x
=
Z
sin
2
x + cos
2
x
cos
3
x
dx =
=
Z
sin
2
x
cos
3
x
+
1
cos x
dx = ln
tg
x
2
+
π
4
−
Z
sin x
cos
3
x
d cos x.
24                                                      §1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .

     Ç) m < 0, |m| = 2k.
                                       Z             k−1
                             dx                1          dx
  Z                   Z
         m
      cos x dx =                   =                            =
                           cos2k x           cos2 x     cos2 x
                                         k−1
                       sin2 x + cos2 x
                 Z                                     Z
             =                                  d tg x = (1 + tg2 x)k−1 d tg x =
                            cos2 x
                                                              Z
                                                           = (1 + t2 )k−1 dt, t = tg x,

ÁÎÁÌÏÇÉÞÎÏ
                                         dx
                 Z                Z                    Z
                       m
                     sin x dx =                =−          (1 + ctg2 x)k−1 d ctg x.
                                       sin2k x
   ðÒÉÍÅÒ 37.

     dx          1     dx        sin2 x + cos2 x
  Z          Z                 Z                         Z
           =                 =                   d tg x = (tg2 x + 1) d tg x =
    cos4 x     cos2 x cos2 x          cos2 x
                                              t3           tg3 x
                               Z
                                  2
                             = (t + 1) dt = + t + C =            + tg x + C.
                                               3             3
   Ä) m < 0, |m| = 2k + 1.

                       dx            cos x
  Z              Z              Z
     cosm x dx =              =              dx =
                    cos2k+1 x      cos2k+2 x
                                                   d sin x               dt
                                            Z                    Z
                                          =                    =
                                               (1 − sin2 x)k+1      (1 − t2 )k+1
                               R dx
ÄÁÌÅÅ ÓÍ. ÐÕÎËÔ 1.4. óÌÕÞÁÊ sin2k+1    x
                                         ÄÅÌÁÅÔÓÑ ÁÎÁÌÏÇÉÞÎÏ.
   ðÒÉÍÅÒ 38.
         dx        cos x dx       d sin x          d sin x            dt
    Z           Z             Z              Z                  Z
              =             =              =                  =               ;
       cos3 x       cos4 x       (cos2 x)2      (1 − sin2 x)2      (1 − t2 )2
(ÄÁÌÅÅ ÓÍ. ÐÕÎËÔ 1.4).
   ÷ ÓÌÕÞÁÅ Ä) ÄÌÑ ÐÏÎÉÖÅÎÉÑ ÓÔÅÐÅÎÉ ÕÄÏÂÎÏ ÐÒÉÍÅÎÑÔØ ÆÏÒÍÕÌÕ sin 2 x +
+ cos2 x = 1.
       dx       sin2 x + cos2 x
  Z           Z
            =                   dx =
     cos3 x          cos3 x
                Z  2              
                     sin x      1               x π       sin x
                                                         Z
              =         3
                            +        dx = ln tg   +    −          d cos x.
                     cos x cos x                 2 4       cos3 x

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