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(sin(2*x)-2*x*cos(2*x))/4еслиx=3 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(2*x) - 2*x*cos(2*x)
-----------------------
           4
$$\frac{- 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{4}$$
Подстановка условия [src]
(sin(2*x) - 2*x*cos(2*x))/4 при x = 3
подставляем
sin(2*x) - 2*x*cos(2*x)
-----------------------
           4
$$\frac{- 2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}}{4}$$
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
переменные
x = 3
$$x = 3$$
sin(2*(3))   (3)*cos(2*(3))
---------- - --------------
    4              2
$$- \frac{(3) \cos{\left(2 (3) \right)}}{2} + \frac{\sin{\left(2 (3) \right)}}{4}$$
sin(2*3)                 
-------- - 1/2*3*cos(2*3)
   4
$$\left(- \frac{1}{2}\right) 3 \cos{\left(2 \cdot 3 \right)} + \frac{\sin{\left(2 \cdot 3 \right)}}{4}$$
  3*cos(6)   sin(6)
- -------- + ------
     2         4
$$- \frac{3 \cos{\left(6 \right)}}{2} + \frac{\sin{\left(6 \right)}}{4}$$
Степени [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
    / -2*I*x    2*I*x\                         
    |e         e     |                         
  x*|------- + ------|     /   -2*I*x    2*I*x\
    \   2        2   /   I*\- e       + e     /
- -------------------- - ----------------------
           2                       8
$$- \frac{x \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)}{2} - \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{8}$$
Численный ответ [src]
0.25*sin(2*x) - 0.5*x*cos(2*x)
Рациональный знаменатель [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
Объединение рациональных выражений [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
Общее упрощение [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
Собрать выражение [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
Общий знаменатель [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
Тригонометрическая часть [src]
/      0        for And(im(x) = 0, 2*x mod pi = 0)                                                  
|                                                                                                   
<   /      pi\                                         //   1      for And(im(x) = 0, x mod pi = 0)\
|cos|2*x - --|              otherwise                x*|<                                          |
\   \      2 /                                         \\cos(2*x)             otherwise            /
-------------------------------------------------- - -----------------------------------------------
                        4                                                   2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{4}\right)$$
/                       0                          for And(im(x) = 0, 2*x mod pi = 0)     //                       1                         for And(im(x) = 0, x mod pi = 0)\
|                                                                                         ||                                                                                 |
|/     0       for And(im(x) = 0, 2*x mod pi = 0)                                         ||/     1        for And(im(x) = 0, x mod pi = 0)                                  |
||                                                                                        |||                                                                                |
<|  2*cot(x)                                                                            x*|<|        2                                                                       |
|<-----------              otherwise                           otherwise                  ||<-1 + cot (x)                                               otherwise            |
||       2                                                                                |||------------             otherwise                                              |
||1 + cot (x)                                                                             |||       2                                                                        |
\\                                                                                        \\\1 + cot (x)                                                                     /
------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------
                                          4                                                                                       2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{4}\right)$$
    1            x     
---------- - ----------
4*csc(2*x)   2*sec(2*x)
$$- \frac{x}{2 \sec{\left(2 x \right)}} + \frac{1}{4 \csc{\left(2 x \right)}}$$
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
   /      pi\             
cos|2*x - --|             
   \      2 /   x*cos(2*x)
------------- - ----------
      4             2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4}$$
                                                  //      1        for And(im(x) = 0, x mod pi = 0)\
                                                  ||                                               |
/   0      for And(im(x) = 0, 2*x mod pi = 0)   x*|<   /pi      \                                  |
<                                                 ||sin|-- + 2*x|             otherwise            |
\sin(2*x)              otherwise                  \\   \2       /                                  /
--------------------------------------------- - ----------------------------------------------------
                      4                                                  2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}}{4}\right)$$
                /pi      \
           x*sin|-- + 2*x|
sin(2*x)        \2       /
-------- - ---------------
   4              2
$$- \frac{x \sin{\left(2 x + \frac{\pi}{2} \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
/      0        for And(im(x) = 0, 2*x mod pi = 0)                                                  
|                                                                                                   
|      1                                               //   1      for And(im(x) = 0, x mod pi = 0)\
<-------------              otherwise                  ||                                          |
|   /      pi\                                       x*|<   1                                      |
|sec|2*x - --|                                         ||--------             otherwise            |
\   \      2 /                                         \\sec(2*x)                                  /
-------------------------------------------------- - -----------------------------------------------
                        4                                                   2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{4}\right)$$
/   0      for And(im(x) = 0, 2*x mod pi = 0)     //   1      for And(im(x) = 0, x mod pi = 0)\
<                                               x*|<                                          |
\sin(2*x)              otherwise                  \\cos(2*x)             otherwise            /
--------------------------------------------- - -----------------------------------------------
                      4                                                2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}}{4}\right)$$
    1               x       
---------- - ---------------
4*csc(2*x)        /pi      \
             2*csc|-- - 2*x|
                  \2       /
$$- \frac{x}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} + \frac{1}{4 \csc{\left(2 x \right)}}$$
/                      0                        for And(im(x) = 0, 2*x mod pi = 0)     //                     1                       for And(im(x) = 0, x mod pi = 0)\
|                                                                                      ||                                                                             |
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{4}\right)$$
/     0       for And(im(x) = 0, 2*x mod pi = 0)     //     1       for And(im(x) = 0, x mod pi = 0)\
|                                                    ||                                             |
|  2*tan(x)                                          ||       2                                     |
<-----------              otherwise                x*|<1 - tan (x)                                  |
|       2                                            ||-----------             otherwise            |
|1 + tan (x)                                         ||       2                                     |
\                                                    \\1 + tan (x)                                  /
------------------------------------------------ - --------------------------------------------------
                       4                                                   2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}}{4}\right)$$
                                                  //      1        for And(im(x) = 0, x mod pi = 0)\
                                                  ||                                               |
/   0      for And(im(x) = 0, 2*x mod pi = 0)     ||      1                                        |
|                                               x*|<-------------             otherwise            |
<   1                                             ||   /pi      \                                  |
|--------              otherwise                  ||csc|-- - 2*x|                                  |
\csc(2*x)                                         \\   \2       /                                  /
--------------------------------------------- - ----------------------------------------------------
                      4                                                  2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}}{4}\right)$$
       1              x     
--------------- - ----------
     /      pi\   2*sec(2*x)
4*sec|2*x - --|             
     \      2 /
$$- \frac{x}{2 \sec{\left(2 x \right)}} + \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
                    /       2   \
     tan(x)       x*\1 - tan (x)/
--------------- - ---------------
  /       2   \     /       2   \
2*\1 + tan (x)/   2*\1 + tan (x)/
$$- \frac{x \left(1 - \tan^{2}{\left(x \right)}\right)}{2 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
/     0       for And(im(x) = 0, 2*x mod pi = 0)     //     1        for And(im(x) = 0, x mod pi = 0)\
|                                                    ||                                              |
|  2*cot(x)                                          ||        2                                     |
<-----------              otherwise                x*|<-1 + cot (x)                                  |
|       2                                            ||------------             otherwise            |
|1 + cot (x)                                         ||       2                                      |
\                                                    \\1 + cot (x)                                   /
------------------------------------------------ - ---------------------------------------------------
                       4                                                    2
$$\left(- \frac{x \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}}{4}\right)$$
Комбинаторика [src]
sin(2*x)   x*cos(2*x)
-------- - ----------
   4           2
$$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
Раскрыть выражение [src]
x   cos(x)*sin(x)        2   
- + ------------- - x*cos (x)
2         2
$$- x \cos^{2}{\left(x \right)} + \frac{x}{2} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2}$$