Калькуляторы онлайн/ Решение уравнений/ Упрощение выражений/ 8*(2*sin(2*x)-24*cos(2*x)/x^3-18*sin(2*x)/x^2+8*cos(2*x)/x+15*sin(2*x)/x^4)/x^2

Общий знаменатель 8*(2*sin(2*x)-24*cos(2*x)/x^3-18*sin(2*x)/x^2+8*cos(2*x)/x+15*sin(2*x)/x^4)/x^2

Выражение, которое надо упростить:

Решение

Вы ввели [src]
  /             24*cos(2*x)   18*sin(2*x)   8*cos(2*x)   15*sin(2*x)\
8*|2*sin(2*x) - ----------- - ----------- + ---------- + -----------|
  |                   3             2           x              4    |
  \                  x             x                          x     /
---------------------------------------------------------------------
                                   2                                 
                                  x
$$\frac{8}{x^{2}} \left(2 \sin{\left (2 x \right )} - \frac{24}{x^{3}} \cos{\left (2 x \right )} - \frac{18}{x^{2}} \sin{\left (2 x \right )} + \frac{8}{x} \cos{\left (2 x \right )} + \frac{15}{x^{4}} \sin{\left (2 x \right )}\right)$$
Численный ответ [src]
8.0*(2.0*sin(2*x) + 8.0*cos(2*x)/x + 15.0*sin(2*x)/x^4 - 24.0*cos(2*x)/x^3 - 18.0*sin(2*x)/x^2)/x^2
Рациональный знаменатель [src]
   4 /  / 2 /                  3         \       3         \      5         \        6         
8*x *\x*\x *\-24*cos(2*x) + 2*x *sin(2*x)/ - 18*x *sin(2*x)/ + 8*x *cos(2*x)/ + 120*x *sin(2*x)
-----------------------------------------------------------------------------------------------
                                               12                                              
                                              x
$$\frac{1}{x^{12}} \left(120 x^{6} \sin{\left (2 x \right )} + 8 x^{4} \left(8 x^{5} \cos{\left (2 x \right )} + x \left(- 18 x^{3} \sin{\left (2 x \right )} + x^{2} \left(2 x^{3} \sin{\left (2 x \right )} - 24 \cos{\left (2 x \right )}\right)\right)\right)\right)$$
Объединение рациональных выражений [src]
  /                  /                3                              2         \\
8*\15*sin(2*x) + 2*x*\-12*cos(2*x) + x *sin(2*x) - 9*x*sin(2*x) + 4*x *cos(2*x)//
---------------------------------------------------------------------------------
                                         6                                       
                                        x
$$\frac{1}{x^{6}} \left(16 x \left(x^{3} \sin{\left (2 x \right )} + 4 x^{2} \cos{\left (2 x \right )} - 9 x \sin{\left (2 x \right )} - 12 \cos{\left (2 x \right )}\right) + 120 \sin{\left (2 x \right )}\right)$$
Общее упрощение [src]
  /                                  2               4               3         \
8*\15*sin(2*x) - 24*x*cos(2*x) - 18*x *sin(2*x) + 2*x *sin(2*x) + 8*x *cos(2*x)/
--------------------------------------------------------------------------------
                                        6                                       
                                       x
$$\frac{1}{x^{6}} \left(16 x^{4} \sin{\left (2 x \right )} + 64 x^{3} \cos{\left (2 x \right )} - 144 x^{2} \sin{\left (2 x \right )} - 192 x \cos{\left (2 x \right )} + 120 \sin{\left (2 x \right )}\right)$$
Собрать выражение [src]
  192*cos(2*x)   144*sin(2*x)   16*sin(2*x)   64*cos(2*x)   120*sin(2*x)
- ------------ - ------------ + ----------- + ----------- + ------------
        5              4              2             3             6     
       x              x              x             x             x
$$\frac{16}{x^{2}} \sin{\left (2 x \right )} + \frac{64}{x^{3}} \cos{\left (2 x \right )} - \frac{144}{x^{4}} \sin{\left (2 x \right )} - \frac{192}{x^{5}} \cos{\left (2 x \right )} + \frac{120}{x^{6}} \sin{\left (2 x \right )}$$ 
              8*18*sin(2*x)   8*24*cos(2*x)   8*8*cos(2*x)   8*15*sin(2*x)
16*sin(2*x) - ------------- - ------------- + ------------ + -------------
                     2               3             x                4     
                    x               x                              x      
--------------------------------------------------------------------------
                                     2                                    
                                    x
$$\frac{1}{x^{2}} \left(16 \sin{\left (2 x \right )} + \frac{120}{x^{4}} \sin{\left (2 x \right )} - \frac{192}{x^{3}} \cos{\left (2 x \right )} - \frac{144}{x^{2}} \sin{\left (2 x \right )} + \frac{64}{x} \cos{\left (2 x \right )}\right)$$
Комбинаторика [src]
  /                                  2               4               3         \
8*\15*sin(2*x) - 24*x*cos(2*x) - 18*x *sin(2*x) + 2*x *sin(2*x) + 8*x *cos(2*x)/
--------------------------------------------------------------------------------
                                        6                                       
                                       x
$$\frac{1}{x^{6}} \left(16 x^{4} \sin{\left (2 x \right )} + 64 x^{3} \cos{\left (2 x \right )} - 144 x^{2} \sin{\left (2 x \right )} - 192 x \cos{\left (2 x \right )} + 120 \sin{\left (2 x \right )}\right)$$
Общий знаменатель [src]
                                     2                4                3         
120*sin(2*x) - 192*x*cos(2*x) - 144*x *sin(2*x) + 16*x *sin(2*x) + 64*x *cos(2*x)
---------------------------------------------------------------------------------
                                         6                                       
                                        x
$$\frac{1}{x^{6}} \left(16 x^{4} \sin{\left (2 x \right )} + 64 x^{3} \cos{\left (2 x \right )} - 144 x^{2} \sin{\left (2 x \right )} - 192 x \cos{\left (2 x \right )} + 120 \sin{\left (2 x \right )}\right)$$
Тригонометрическая часть [src]
              192*cos(2*x)   144*sin(2*x)   64*cos(2*x)   120*sin(2*x)
16*sin(2*x) - ------------ - ------------ + ----------- + ------------
                    3              2             x              4     
                   x              x                            x      
----------------------------------------------------------------------
                                   2                                  
                                  x
$$\frac{1}{x^{2}} \left(16 \sin{\left (2 x \right )} + \frac{64}{x} \cos{\left (2 x \right )} - \frac{144}{x^{2}} \sin{\left (2 x \right )} - \frac{192}{x^{3}} \cos{\left (2 x \right )} + \frac{120}{x^{4}} \sin{\left (2 x \right )}\right)$$
Раскрыть выражение [src]
    /        2            2   \     /       2           2   \                                                           
  8*\- 24*sin (x) + 24*cos (x)/   8*\- 8*sin (x) + 8*cos (x)/                      288*cos(x)*sin(x)   240*cos(x)*sin(x)
- ----------------------------- + --------------------------- + 32*cos(x)*sin(x) - ----------------- + -----------------
                 3                             x                                            2                   4       
                x                                                                          x                   x        
------------------------------------------------------------------------------------------------------------------------
                                                            2                                                           
                                                           x
$$\frac{1}{x^{2}} \left(32 \sin{\left (x \right )} \cos{\left (x \right )} + \frac{1}{x} \left(- 64 \sin^{2}{\left (x \right )} + 64 \cos^{2}{\left (x \right )}\right) - \frac{288}{x^{2}} \sin{\left (x \right )} \cos{\left (x \right )} - \frac{1}{x^{3}} \left(- 192 \sin^{2}{\left (x \right )} + 192 \cos^{2}{\left (x \right )}\right) + \frac{240}{x^{4}} \sin{\left (x \right )} \cos{\left (x \right )}\right)$$
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