tan(3*x)^5+x*tan(3*x)^4*( ... +15*tan(3*x)^2) если x=-1 (упростите выражение)

Вы ввели

   5             4      /           2     \
tan (3*x) + x*tan (3*x)*\15 + 15*tan (3*x)/

$$x \tan^{4}{\left (3 x \right )} \left(15 \tan^{2}{\left (3 x \right )} + 15\right) + \tan^{5}{\left (3 x \right )}$$

Подстановка условия

tan(3*x)^5 + (x*tan(3*x)^4)*(15 + 15*tan(3*x)^2) при x = -1

tan(3*x)^5 + (x*tan(3*x)^4)*(15 + 15*tan(3*x)^2)

$$x \tan^{4}{\left (3 x \right )} \left(15 \tan^{2}{\left (3 x \right )} + 15\right) + \tan^{5}{\left (3 x \right )}$$

tan(3*(-1))^5 + ((-1)*tan(3*(-1))^4)*(15 + 15*tan(3*(-1))^2)

$$(-1) \tan^{4}{\left (3 (-1) \right )} \left(15 \tan^{2}{\left (3 (-1) \right )} + 15\right) + \tan^{5}{\left (3 (-1) \right )}$$

tan(3*(-1))^5 + (-tan(3*(-1))^4)*(15 + 15*tan(3*(-1))^2)

$$\left(15 \tan^{2}{\left (-1 \cdot 3 \right )} + 15\right) \left(- \tan^{4}{\left (-1 \cdot 3 \right )}\right) + \tan^{5}{\left (-1 \cdot 3 \right )}$$

-tan(3)^5 - tan(3)^4*(15 + 15*tan(3)^2)

$$- \left(15 \tan^{2}{\left (3 \right )} + 15\right) \tan^{4}{\left (3 \right )} - \tan^{5}{\left (3 \right )}$$

Степени

   5             4      /           2     \
tan (3*x) + x*tan (3*x)*\15 + 15*tan (3*x)/

$$x \left(15 \tan^{2}{\left (3 x \right )} + 15\right) \tan^{4}{\left (3 x \right )} + \tan^{5}{\left (3 x \right )}$$

Численный ответ

tan(3*x)^5 + x*tan(3*x)^4*(15.0 + 15.0*tan(3*x)^2)

Рациональный знаменатель

   5             4      /           2     \
tan (3*x) + x*tan (3*x)*\15 + 15*tan (3*x)/

$$x \left(15 \tan^{2}{\left (3 x \right )} + 15\right) \tan^{4}{\left (3 x \right )} + \tan^{5}{\left (3 x \right )}$$

Объединение рациональных выражений

   4      /     /       2     \           \
tan (3*x)*\15*x*\1 + tan (3*x)/ + tan(3*x)/

$$\left(15 x \left(\tan^{2}{\left (3 x \right )} + 1\right) + \tan{\left (3 x \right )}\right) \tan^{4}{\left (3 x \right )}$$

Общее упрощение

   4      /   15*x             \
tan (3*x)*|--------- + tan(3*x)|
          |   2                |
          \cos (3*x)           /

$$\left(\frac{15 x}{\cos^{2}{\left (3 x \right )}} + \tan{\left (3 x \right )}\right) \tan^{4}{\left (3 x \right )}$$

Собрать выражение

   5                2         4     
tan (3*x) + 15*x*sec (3*x)*tan (3*x)

$$15 x \tan^{4}{\left (3 x \right )} \sec^{2}{\left (3 x \right )} + \tan^{5}{\left (3 x \right )}$$

Общий знаменатель

   5                4                6     
tan (3*x) + 15*x*tan (3*x) + 15*x*tan (3*x)

$$15 x \tan^{6}{\left (3 x \right )} + 15 x \tan^{4}{\left (3 x \right )} + \tan^{5}{\left (3 x \right )}$$

Тригонометрическая часть

                    4     
   5        15*x*tan (3*x)
tan (3*x) + --------------
                 2        
              cos (3*x)   

$$\frac{15 x \tan^{4}{\left (3 x \right )}}{\cos^{2}{\left (3 x \right )}} + \tan^{5}{\left (3 x \right )}$$

Комбинаторика

   4      /               2                \
tan (3*x)*\15*x + 15*x*tan (3*x) + tan(3*x)/

$$\left(15 x \tan^{2}{\left (3 x \right )} + 15 x + \tan{\left (3 x \right )}\right) \tan^{4}{\left (3 x \right )}$$

Раскрыть выражение

                                                    /                              2\
                                                  4 |        /     3              \ |
                            /     3              \  |     15*\- tan (x) + 3*tan(x)/ |
                          x*\- tan (x) + 3*tan(x)/ *|15 + --------------------------|
                      5                             |                         2     |
/     3              \                              |          /         2   \      |
\- tan (x) + 3*tan(x)/                              \          \1 - 3*tan (x)/      /
----------------------- + -----------------------------------------------------------
                   5                                           4                     
    /         2   \                             /         2   \                      
    \1 - 3*tan (x)/                             \1 - 3*tan (x)/                      

$$\frac{x \left(- \tan^{3}{\left (x \right )} + 3 \tan{\left (x \right )}\right)^{4}}{\left(- 3 \tan^{2}{\left (x \right )} + 1\right)^{4}} \left(15 + \frac{15 \left(- \tan^{3}{\left (x \right )} + 3 \tan{\left (x \right )}\right)^{2}}{\left(- 3 \tan^{2}{\left (x \right )} + 1\right)^{2}}\right) + \frac{\left(- \tan^{3}{\left (x \right )} + 3 \tan{\left (x \right )}\right)^{5}}{\left(- 3 \tan^{2}{\left (x \right )} + 1\right)^{5}}$$