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Общий знаменатель -log(((2*sin(x))/(1+cos(x ... (x))/(1+cos(x))))/sqrt(2)

Учитель очень удивится увидев твоё верное решение 😼

Примеры

Решение

Вы ввели [src]
    / 2*sin(x)     3/2     \ 
    |---------- - 2    - 2 | 
    |1 + cos(x)            | 
-log|----------------------| 
    |      3/2    2*sin(x) | 
    |-2 + 2    + ----------| 
    \            1 + cos(x)/ 
-----------------------------
              ___            
            \/ 2
$$\frac{1}{\sqrt{2}} \left(-1 \log{\left (\frac{- 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} - 2}{-2 + 2^{\frac{3}{2}} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}\right)$$
Степени [src]
          /         ___    2*sin(x) \ 
          |-2 - 2*\/ 2  + ----------| 
   ___    |               1 + cos(x)| 
-\/ 2 *log|-------------------------| 
          |         ___    2*sin(x) | 
          |-2 + 2*\/ 2  + ----------| 
          \               1 + cos(x)/ 
--------------------------------------
                  2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{- 2 \sqrt{2} - 2 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{-2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Численный ответ [src]
-0.707106781186547*log(((2*sin(x))/(1 + cos(x)) - 2^(3/2) - 2)/(-2 + 2^(3/2) + (2*sin(x))/(1 + cos(x))))
Рациональный знаменатель [src]
          /   /                    ___     ___ /      3/2                  ___           2*sin(x)    2*cos(x)*sin(x)\   /        ___    2*sin(x) \              ___           2*sin(x)    2*cos(x)*sin(x)\\ 
          |-2*|-2 - 2*cos(x) + 2*\/ 2  + \/ 2 *|-2 + 2    - 2*cos(x) + 2*\/ 2 *cos(x) + ---------- + ---------------| + |2 - 2*\/ 2  - ----------|*sin(x) + 2*\/ 2 *cos(x) + ---------- + ---------------|| 
   ___    |   \                                \                                        1 + cos(x)      1 + cos(x)  /   \              1 + cos(x)/                           1 + cos(x)      1 + cos(x)  /| 
-\/ 2 *log|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| 
          |                                               /         ___    2*sin(x) \ /      3/2                  ___           2*sin(x)    2*cos(x)*sin(x)\                                              | 
          |                                               |-2 + 2*\/ 2  + ----------|*|-2 + 2    - 2*cos(x) + 2*\/ 2 *cos(x) + ---------- + ---------------|                                              | 
          \                                               \               1 + cos(x)/ \                                        1 + cos(x)      1 + cos(x)  /                                              / 
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                     2
$$- \frac{\sqrt{2}}{2} \log{\left (- \frac{2 \left(- 2 \sqrt{2} + 2 - \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) \sin{\left (x \right )} + 2 \sqrt{2} \left(- 2 \cos{\left (x \right )} + 2 \sqrt{2} \cos{\left (x \right )} - 2 + 2^{\frac{3}{2}} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) - 4 \cos{\left (x \right )} + 4 \sqrt{2} \cos{\left (x \right )} - 4 + 4 \sqrt{2} + \frac{4 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{4 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{\left(-2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) \left(- 2 \cos{\left (x \right )} + 2 \sqrt{2} \cos{\left (x \right )} - 2 + 2^{\frac{3}{2}} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right)} \right )}$$
Объединение рациональных выражений [src]
          /                ___                      \ 
   ___    |-1 - cos(x) - \/ 2 *(1 + cos(x)) + sin(x)| 
-\/ 2 *log|-----------------------------------------| 
          |                 /       ___\            | 
          \    (1 + cos(x))*\-1 + \/ 2 / + sin(x)   / 
------------------------------------------------------
                          2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{- \sqrt{2} \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )} - \cos{\left (x \right )} - 1}{\left(-1 + \sqrt{2}\right) \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )}} \right )}$$
Общее упрощение [src]
          /          /      ___\             \ 
   ___    |-sin(x) + \1 + \/ 2 /*(1 + cos(x))| 
-\/ 2 *log|----------------------------------| 
          |          /      ___\             | 
          \-sin(x) + \1 - \/ 2 /*(1 + cos(x))/ 
-----------------------------------------------
                       2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{\left(1 + \sqrt{2}\right) \left(\cos{\left (x \right )} + 1\right) - \sin{\left (x \right )}}{\left(- \sqrt{2} + 1\right) \left(\cos{\left (x \right )} + 1\right) - \sin{\left (x \right )}} \right )}$$
Собрать выражение [src]
          / 2*sin(x)     3/2     \ 
          |---------- - 2    - 2 | 
   ___    |1 + cos(x)            | 
-\/ 2 *log|----------------------| 
          |      3/2    2*sin(x) | 
          |-2 + 2    + ----------| 
          \            1 + cos(x)/ 
-----------------------------------
                 2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{- 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} - 2}{-2 + 2^{\frac{3}{2}} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
         /         ___              /         ___\       \
  ___    |-2 + 2*\/ 2  + 2*sin(x) + \-2 + 2*\/ 2 /*cos(x)|
\/ 2 *log|-----------------------------------------------|
         |         ___              /         ___\       |
         \-2 - 2*\/ 2  + 2*sin(x) + \-2 - 2*\/ 2 /*cos(x)/
----------------------------------------------------------
                            2
$$\frac{\sqrt{2}}{2} \log{\left (\frac{2 \sin{\left (x \right )} + \left(-2 + 2 \sqrt{2}\right) \cos{\left (x \right )} - 2 + 2 \sqrt{2}}{2 \sin{\left (x \right )} + \left(- 2 \sqrt{2} - 2\right) \cos{\left (x \right )} - 2 \sqrt{2} - 2} \right )}$$
Общий знаменатель [src]
          /                                                                                                         ___         \ 
   ___    |             1                                           sin(x)                                        \/ 2          | 
-\/ 2 *log|- ----------------------- + --------------------------------------------------------------- - -----------------------| 
          |         ___     sin(x)            ___              ___            sin(x)     cos(x)*sin(x)          ___     sin(x)  | 
          |  -1 + \/ 2  + ----------   -1 + \/ 2  - cos(x) + \/ 2 *cos(x) + ---------- + -------------   -1 + \/ 2  + ----------| 
          \               1 + cos(x)                                        1 + cos(x)     1 + cos(x)                 1 + cos(x)/ 
----------------------------------------------------------------------------------------------------------------------------------
                                                                2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{\sin{\left (x \right )}}{- \cos{\left (x \right )} + \sqrt{2} \cos{\left (x \right )} - 1 + \sqrt{2} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{\sqrt{2}}{-1 + \sqrt{2} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{1}{-1 + \sqrt{2} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Тригонометрическая часть [src]
          /         ___    2*sin(x) \ 
          |-2 - 2*\/ 2  + ----------| 
   ___    |               1 + cos(x)| 
-\/ 2 *log|-------------------------| 
          |        3/2    2*sin(x)  | 
          |  -2 + 2    + ---------- | 
          \              1 + cos(x) / 
--------------------------------------
                  2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{- 2 \sqrt{2} - 2 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{-2 + 2^{\frac{3}{2}} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Комбинаторика [src]
          /                                           ___                                                                                   \ 
   ___    |              2                        2*\/ 2                                             2*sin(x)                               | 
-\/ 2 *log|- ------------------------- - ------------------------- + -----------------------------------------------------------------------| 
          |           ___    2*sin(x)             ___    2*sin(x)                        ___       ___           2*sin(x)    2*cos(x)*sin(x)| 
          |  -2 + 2*\/ 2  + ----------   -2 + 2*\/ 2  + ----------   -2 - 2*cos(x) + 2*\/ 2  + 2*\/ 2 *cos(x) + ---------- + ---------------| 
          \                 1 + cos(x)                  1 + cos(x)                                              1 + cos(x)      1 + cos(x)  / 
----------------------------------------------------------------------------------------------------------------------------------------------
                                                                      2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{2 \sin{\left (x \right )}}{- 2 \cos{\left (x \right )} + 2 \sqrt{2} \cos{\left (x \right )} - 2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{2 \sqrt{2}}{-2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{2}{-2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Раскрыть выражение [src]
  ___ /     /         ___    2*sin(x) \      /         ___    2*sin(x) \\
\/ 2 *|- log|-2 - 2*\/ 2  + ----------| + log|-2 + 2*\/ 2  + ----------||
      \     \               1 + cos(x)/      \               1 + cos(x)//
-------------------------------------------------------------------------
                                    2
$$\frac{\sqrt{2}}{2} \left(\log{\left (-2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \right )} - \log{\left (- 2 \sqrt{2} - 2 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \right )}\right)$$
          /         ___    2*sin(x) \ 
          |-2 - 2*\/ 2  + ----------| 
   ___    |               1 + cos(x)| 
-\/ 2 *log|-------------------------| 
          |         ___    2*sin(x) | 
          |-2 + 2*\/ 2  + ----------| 
          \               1 + cos(x)/ 
--------------------------------------
                  2
$$- \frac{\sqrt{2}}{2} \log{\left (\frac{- 2 \sqrt{2} - 2 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{-2 + 2 \sqrt{2} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$