Сборник задач по высшей математике. Часть IV. Интегралы. Дифференциальные уравнения. Самохин А.В - 10 стр.

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МГТУ ГА | Москва

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10 çÌÁ×Á I. éÎÔÅÇÒÁÌØÎÏÅ ÉÓÞÉÓÌÅÎÉÅ ÆÕÎËÃÉÉ ÏÄÎÏÊ ÐÅÒÅÍÅÎÎÏÊ
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.
10          çÌÁ×Á I. éÎÔÅÇÒÁÌØÎÏÅ ÉÓÞÉÓÌÅÎÉÅ ÆÕÎËÃÉÉ ÏÄÎÏÊ ÐÅÒÅÍÅÎÎÏÊ

     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
           3                3             3
        sin x cos x dx = sin x d sin x = t dt = + C =        + 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
               =                dx =                 =
   sin x cos x     sin x cos2 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

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