(4*sin(10*x)-40*x*cos(10*x)-25*sin(4*x)+100*x*cos(4*x))*1/800 если x=1 (упростите выражение)

Выражение, которое надо упростить:
Например, 1/(a*x-1)-1/(a*x+1)

    Решение

    Вы ввели
    [LaTeX]
    4*sin(10*x) - 40*x*cos(10*x) - 25*sin(4*x) + 100*x*cos(4*x)
    -----------------------------------------------------------
                                800                            
    $$\frac{1}{800} \left(100 x \cos{\left (4 x \right )} + - 40 x \cos{\left (10 x \right )} + 4 \sin{\left (10 x \right )} - 25 \sin{\left (4 x \right )}\right)$$
    Подстановка условия
    [LaTeX]
    (4*sin(10*x) - 40*x*cos(10*x) - 25*sin(4*x) + (100*x)*cos(4*x))/800 при x = 1
    (4*sin(10*x) - 40*x*cos(10*x) - 25*sin(4*x) + (100*x)*cos(4*x))/800
    $$\frac{1}{800} \left(100 x \cos{\left (4 x \right )} + - 40 x \cos{\left (10 x \right )} + 4 \sin{\left (10 x \right )} - 25 \sin{\left (4 x \right )}\right)$$
    (4*sin(10*(1)) - 40*(1)*cos(10*(1)) - 25*sin(4*(1)) + (100*(1))*cos(4*(1)))/800
    $$\frac{1}{800} \left(100 (1) \cos{\left (4 (1) \right )} + - 40 (1) \cos{\left (10 (1) \right )} + 4 \sin{\left (10 (1) \right )} - 25 \sin{\left (4 (1) \right )}\right)$$
    (4*sin(10) - 40*cos(10) - 25*sin(4) + 100*cos(4))/800
    $$\frac{1}{800} \left(100 \cos{\left (4 \right )} + - 25 \sin{\left (4 \right )} + 4 \sin{\left (10 \right )} - 40 \cos{\left (10 \right )}\right)$$
    -cos(10)/20 - sin(4)/32 + cos(4)/8 + sin(10)/200
    $$\frac{1}{8} \cos{\left (4 \right )} + \frac{1}{200} \sin{\left (10 \right )} - \frac{1}{32} \sin{\left (4 \right )} - \frac{1}{20} \cos{\left (10 \right )}$$
    Степени
    [LaTeX]
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Численный ответ
    [LaTeX]
    0.005*sin(10*x) - 0.03125*sin(4*x) + 0.125*x*cos(4*x) - 0.05*x*cos(10*x)
    Рациональный знаменатель
    [LaTeX]
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Объединение рациональных выражений
    [LaTeX]
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Общее упрощение
    [LaTeX]
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Собрать выражение
    [LaTeX]
    sin(10*x)   25*sin(4*x)   40*x*cos(10*x)   100*x*cos(4*x)
    --------- - ----------- - -------------- + --------------
       200          800            800              800      
    $$- \frac{x}{20} \cos{\left (10 x \right )} + \frac{100 x}{800} \cos{\left (4 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Общий знаменатель
    [LaTeX]
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Тригонометрическая часть
    [LaTeX]
      sin(4*x)   sin(10*x)   x*cos(10*x)   x*cos(4*x)
    - -------- + --------- - ----------- + ----------
         32         200           20           8     
    $$\frac{x}{8} \cos{\left (4 x \right )} - \frac{x}{20} \cos{\left (10 x \right )} - \frac{1}{32} \sin{\left (4 x \right )} + \frac{1}{200} \sin{\left (10 x \right )}$$
    Комбинаторика
    [LaTeX]
    -(-4*sin(10*x) + 25*sin(4*x) - 100*x*cos(4*x) + 40*x*cos(10*x)) 
    ----------------------------------------------------------------
                                  800                               
    $$- \frac{1}{800} \left(- 100 x \cos{\left (4 x \right )} + 40 x \cos{\left (10 x \right )} + 25 \sin{\left (4 x \right )} - 4 \sin{\left (10 x \right )}\right)$$
    Раскрыть выражение
    [LaTeX]
           3       7           7       3         3               /   10         10             4       6            8       2            2       8             6       4   \     /   4         4           2       2   \      3                9                9                   5       5   
      3*cos (x)*sin (x)   3*cos (x)*sin (x)   cos (x)*sin(x)   x*\cos  (x) - sin  (x) - 210*cos (x)*sin (x) - 45*cos (x)*sin (x) + 45*cos (x)*sin (x) + 210*cos (x)*sin (x)/   x*\cos (x) + sin (x) - 6*cos (x)*sin (x)/   sin (x)*cos(x)   cos (x)*sin(x)   sin (x)*cos(x)   63*cos (x)*sin (x)
    - ----------------- - ----------------- - -------------- - ------------------------------------------------------------------------------------------------------------- + ----------------------------------------- + -------------- + -------------- + -------------- + ------------------
              5                   5                 8                                                                20                                                                            8                             8                20               20                 50        
    $$\frac{x}{8} \left(\sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}\right) - \frac{x}{20} \left(- \sin^{10}{\left (x \right )} + 45 \sin^{8}{\left (x \right )} \cos^{2}{\left (x \right )} - 210 \sin^{6}{\left (x \right )} \cos^{4}{\left (x \right )} + 210 \sin^{4}{\left (x \right )} \cos^{6}{\left (x \right )} - 45 \sin^{2}{\left (x \right )} \cos^{8}{\left (x \right )} + \cos^{10}{\left (x \right )}\right) + \frac{1}{20} \sin^{9}{\left (x \right )} \cos{\left (x \right )} - \frac{3}{5} \sin^{7}{\left (x \right )} \cos^{3}{\left (x \right )} + \frac{63}{50} \sin^{5}{\left (x \right )} \cos^{5}{\left (x \right )} - \frac{3}{5} \sin^{3}{\left (x \right )} \cos^{7}{\left (x \right )} + \frac{1}{8} \sin^{3}{\left (x \right )} \cos{\left (x \right )} + \frac{1}{20} \sin{\left (x \right )} \cos^{9}{\left (x \right )} - \frac{1}{8} \sin{\left (x \right )} \cos^{3}{\left (x \right )}$$