Интегральное исчисление функции одной переменной. - 5 стр.

UptoLike

ВУЗ: 

МГТУ ГА | Москва

Рубрика: 

§1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . . 5
ðÒÉÍÅÒ 4.
Z
dx
√
5x + 2
=
1
5
Z
d(5x + 2)
√
5x + 2
=
=
1
5
Z
dt
√
t
=
1
5
Z
t
−1/2
dt =
2
5
t
1/2
+ C =
2
5
√
5x + 2 + C.
ðÒÉÍÅÒ 5.
Z
sin(3x −4) dx =
1
3
Z
sin(3x −4) d(3x − 4) =
=
1
3
Z
sin t dt = −
1
3
cos t + C = −
1
3
cos(3x − 4) + C.
ðÒÉÍÅÒ 6.
Z
dx
2x − 6
=
1
2
Z
d(2x − 6)
2x − 6
=
1
2
Z
dt
t
=
1
2
ln |t| + C =
1
2
ln |2x − 6| + C.
ðÒÉÍÅÒ 7.
Z
sin
3
x cos x dx =
Z
sin
3
x d sin x =
Z
t
3
dt =
t
4
4
+ C =
sin
4
x
4
+ C.
ðÒÉÍÅÒ 8.
Z
sin x
4 − cos x
dx =
Z
d(−cos x)
4 − cos x
=
Z
d(−cos x + 4)
4 − cos x
=
=
Z
dt
t
= ln |t| + C = ln |4 − cos x| + C.
ðÒÉÍÅÒ 9.
Z
dx
sin x cos x
=
Z
cos x
sin x cos
2
x
dx =
Z
1
tg x
dx
cos
2
x
=
=
Z
d tg x
tg x
=
Z
dt
t
= ln |t| + C = ln |tg x|+ C.
ðÒÉÍÅÒ 10.
Z
ln x
x
dx =
Z
ln x d ln x =
Z
t dt =
t
2
2
+ C =
1
2
ln
2
x + C.
§1. îÅÏÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ. . .                                                      5

  ðÒÉÍÅÒ 4.

       dx     1       d(5x + 2)
 Z                Z
     √      =          √        =
      5x + 2 5          5x + 2
                        1   dt    1             2           2√
                          Z         Z
                      =     √ =       t−1/2 dt = t1/2 + C =   5x + 2 + C.
                        5     t 5               5           5
  ðÒÉÍÅÒ 5.

                    1
 Z                    Z
   sin(3x − 4) dx =     sin(3x − 4) d(3x − 4) =
                    3
                        1                1             1
                          Z
                      =      sin t dt = − cos t + C = − cos(3x − 4) + C.
                        3                3             3
  ðÒÉÍÅÒ 6.
      dx    1 d(2x − 6) 1 dt 1               1
  Z          Z             Z
          =            =       = ln |t| + C = ln |2x − 6| + C.
    2x − 6 2   2x − 6    2   t  2            2
  ðÒÉÍÅÒ 7.

                                             t4     sin4 x
   Z                 Z               Z
     sin x cos x dx = sin x d sin x = t3 dt = + C =
        3                3
                                                           + C.
                                             4        4
  ðÒÉÍÅÒ 8.

       sin x              d(− cos x)       d(− cos x + 4)
 Z                    Z                  Z
               dx =                  =                     =
     4 − cos x            4 − cos x          4 − cos x
                                              dt
                                           Z
                                         =       = ln |t| + C = ln |4 − cos x| + C.
                                               t
  ðÒÉÍÅÒ 9.

       dx             cos x               1    dx
 Z               Z                    Z
               =            2
                                dx =                 =
   sin x cos x     sin x cos x           tg x cos2 x
                                   d tg x       dt
                                Z            Z
                              =           =        = ln |t| + C = ln | tg x| + C.
                                    tg x         t
  ðÒÉÍÅÒ 10.
         ln x                          t2    1
       Z          Z             Z
              dx = ln x d ln x = t dt = + C = ln2 x + C.
          x                            2     2

Страницы