Господин Экзамен
Упрощение выражений/
5*exp(5*z)/(sin(z)+z^3-z)+(e^(5*z)-1)*(1-cos(z)-3*z^2)/((sin(z)+z^3-z)^2) если z=3
5*exp(5*z)/(sin(z)+z^3-z)+(e^(5*z)-1)*(1-cos(z)-3*z^2)/((sin(z)+z^3-z)^2) если z=3
Решение
Вы ввели
[src]
5*z / 5*z \ / 2\
5*e \e - 1/*\1 - cos(z) - 3*z /
--------------- + ------------------------------
3 2
sin(z) + z - z / 3 \
\sin(z) + z - z/
$$\frac{5 e^{5 z}}{z^{3} - z + \sin{\left(z \right)}} + \frac{\left(e^{5 z} - 1\right) \left(- 3 z^{2} - \cos{\left(z \right)} + 1\right)}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
5*exp(5*z)/(sin(z) + z^3 - z) + (E^(5*z) - 1*1)*(1 - cos(z) - 3*z^2)/((sin(z) + z^3 - z)^2)
Общее упрощение
[src]
/ 5*z\ / 2 \ / 3 \ 5*z
\1 - e /*\-1 + 3*z + cos(z)/ + 5*\z - z + sin(z)/*e
----------------------------------------------------------
2
/ 3 \
\z - z + sin(z)/
$$\frac{5 \left(z^{3} - z + \sin{\left(z \right)}\right) e^{5 z} + \left(- e^{5 z} + 1\right) \left(3 z^{2} + \cos{\left(z \right)} - 1\right)}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
((1 - exp(5*z))*(-1 + 3*z^2 + cos(z)) + 5*(z^3 - z + sin(z))*exp(5*z))/(z^3 - z + sin(z))^2
Подстановка условия
[src]
5*exp(5*z)/(sin(z) + z^3 - z) + (E^(5*z) - 1*1)*(1 - cos(z) - 3*z^2)/((sin(z) + z^3 - z)^2) при z = 3
подставляем
5*z / 5*z \ / 2\
5*e \e - 1/*\1 - cos(z) - 3*z /
--------------- + ------------------------------
3 2
sin(z) + z - z / 3 \
\sin(z) + z - z/
$$\frac{5 e^{5 z}}{z^{3} - z + \sin{\left(z \right)}} + \frac{\left(e^{5 z} - 1\right) \left(- 3 z^{2} - \cos{\left(z \right)} + 1\right)}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
/ 5*z\ / 2 \ / 3 \ 5*z
\1 - e /*\-1 + 3*z + cos(z)/ + 5*\z - z + sin(z)/*e
----------------------------------------------------------
2
/ 3 \
\z - z + sin(z)/
$$\frac{5 \left(z^{3} - z + \sin{\left(z \right)}\right) e^{5 z} + \left(- e^{5 z} + 1\right) \left(3 z^{2} + \cos{\left(z \right)} - 1\right)}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
переменные
z = 3
$$z = 3$$
/ 5*(3)\ / 2 \ / 3 \ 5*(3)
\1 - e /*\-1 + 3*(3) + cos((3))/ + 5*\(3) - (3) + sin((3))/*e
------------------------------------------------------------------------
2
/ 3 \
\(3) - (3) + sin((3))/
$$\frac{5 \left((3)^{3} - (3) + \sin{\left((3) \right)}\right) e^{5 (3)} + \left(- e^{5 (3)} + 1\right) \left(3 (3)^{2} + \cos{\left((3) \right)} - 1\right)}{\left((3)^{3} - (3) + \sin{\left((3) \right)}\right)^{2}}$$
/ 5*3\ / 2 \ / 3 \ 5*3
\1 - e /*\-1 + 3*3 + cos(3)/ + 5*\3 - 3 + sin(3)/*e
----------------------------------------------------------
2
/ 3 \
\3 - 3 + sin(3)/
$$\frac{\left(- e^{5 \cdot 3} + 1\right) \left(-1 + \cos{\left(3 \right)} + 3 \cdot 3^{2}\right) + 5 \left(\left(-1\right) 3 + \sin{\left(3 \right)} + 3^{3}\right) e^{5 \cdot 3}}{\left(\left(-1\right) 3 + \sin{\left(3 \right)} + 3^{3}\right)^{2}}$$
/ 15\ 15
\1 - e /*(26 + cos(3)) + 5*(24 + sin(3))*e
---------------------------------------------
2
(24 + sin(3))
$$\frac{\left(- e^{15} + 1\right) \left(\cos{\left(3 \right)} + 26\right) + 5 \left(\sin{\left(3 \right)} + 24\right) e^{15}}{\left(\sin{\left(3 \right)} + 24\right)^{2}}$$
((1 - exp(15))*(26 + cos(3)) + 5*(24 + sin(3))*exp(15))/(24 + sin(3))^2
Собрать выражение
[src]
2 5*z 5*z 5*z 2 5*z
2 6*z 2*cos(z) 2*e 5*e 2*cos(z)*e 6*z *e
------------------------------------------------------------- - ------------------------------------------------------------- - ------------------------------------------------------------- - ------------------------------------------------------------- + --------------- + ------------------------------------------------------------- + -------------------------------------------------------------
2 6 4 3 2 6 4 3 2 6 4 3 2 6 4 3 3 2 6 4 3 2 6 4 3
-1 - 2*z - 2*z + 4*z - 4*z *sin(z) + 4*z*sin(z) + cos(2*z) -1 - 2*z - 2*z + 4*z - 4*z *sin(z) + 4*z*sin(z) + cos(2*z) -1 - 2*z - 2*z + 4*z - 4*z *sin(z) + 4*z*sin(z) + cos(2*z) -1 - 2*z - 2*z + 4*z - 4*z *sin(z) + 4*z*sin(z) + cos(2*z) z - z + sin(z) -1 - 2*z - 2*z + 4*z - 4*z *sin(z) + 4*z*sin(z) + cos(2*z) -1 - 2*z - 2*z + 4*z - 4*z *sin(z) + 4*z*sin(z) + cos(2*z)
$$\frac{6 z^{2} e^{5 z}}{- 2 z^{6} + 4 z^{4} - 4 z^{3} \sin{\left(z \right)} - 2 z^{2} + 4 z \sin{\left(z \right)} + \cos{\left(2 z \right)} - 1} + \frac{2 e^{5 z} \cos{\left(z \right)}}{- 2 z^{6} + 4 z^{4} - 4 z^{3} \sin{\left(z \right)} - 2 z^{2} + 4 z \sin{\left(z \right)} + \cos{\left(2 z \right)} - 1} - \frac{2 e^{5 z}}{- 2 z^{6} + 4 z^{4} - 4 z^{3} \sin{\left(z \right)} - 2 z^{2} + 4 z \sin{\left(z \right)} + \cos{\left(2 z \right)} - 1} + \frac{5 e^{5 z}}{z^{3} - z + \sin{\left(z \right)}} - \frac{6 z^{2}}{- 2 z^{6} + 4 z^{4} - 4 z^{3} \sin{\left(z \right)} - 2 z^{2} + 4 z \sin{\left(z \right)} + \cos{\left(2 z \right)} - 1} - \frac{2 \cos{\left(z \right)}}{- 2 z^{6} + 4 z^{4} - 4 z^{3} \sin{\left(z \right)} - 2 z^{2} + 4 z \sin{\left(z \right)} + \cos{\left(2 z \right)} - 1} + \frac{2}{- 2 z^{6} + 4 z^{4} - 4 z^{3} \sin{\left(z \right)} - 2 z^{2} + 4 z \sin{\left(z \right)} + \cos{\left(2 z \right)} - 1}$$
2/(-1 - 2*z^2 - 2*z^6 + 4*z^4 - 4*z^3*sin(z) + 4*z*sin(z) + cos(2*z)) - 6*z^2/(-1 - 2*z^2 - 2*z^6 + 4*z^4 - 4*z^3*sin(z) + 4*z*sin(z) + cos(2*z)) - 2*cos(z)/(-1 - 2*z^2 - 2*z^6 + 4*z^4 - 4*z^3*sin(z) + 4*z*sin(z) + cos(2*z)) - 2*exp(5*z)/(-1 - 2*z^2 - 2*z^6 + 4*z^4 - 4*z^3*sin(z) + 4*z*sin(z) + cos(2*z)) + 5*exp(5*z)/(z^3 - z + sin(z)) + 2*cos(z)*exp(5*z)/(-1 - 2*z^2 - 2*z^6 + 4*z^4 - 4*z^3*sin(z) + 4*z*sin(z) + cos(2*z)) + 6*z^2*exp(5*z)/(-1 - 2*z^2 - 2*z^6 + 4*z^4 - 4*z^3*sin(z) + 4*z*sin(z) + cos(2*z))
Комбинаторика
[src]
2 5*z 5*z 2 5*z 3 5*z 5*z 5*z
-1 + 3*z - cos(z)*e - 5*z*e - 3*z *e + 5*z *e + 5*e *sin(z) + cos(z) + e
------------------------------------------------------------------------------------------
2
/ 3 \
\z - z + sin(z)/
$$\frac{5 z^{3} e^{5 z} - 3 z^{2} e^{5 z} - 5 z e^{5 z} + 5 e^{5 z} \sin{\left(z \right)} - e^{5 z} \cos{\left(z \right)} + e^{5 z} + 3 z^{2} + \cos{\left(z \right)} - 1}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
(-1 + 3*z^2 - cos(z)*exp(5*z) - 5*z*exp(5*z) - 3*z^2*exp(5*z) + 5*z^3*exp(5*z) + 5*exp(5*z)*sin(z) + cos(z) + exp(5*z))/(z^3 - z + sin(z))^2
Объединение рациональных выражений
[src]
/ 5*z\ / 2\ / 3 \ 5*z
\-1 + e /*\1 - cos(z) - 3*z / + 5*\z - z + sin(z)/*e
----------------------------------------------------------
2
/ 3 \
\z - z + sin(z)/
$$\frac{5 \left(z^{3} - z + \sin{\left(z \right)}\right) e^{5 z} + \left(e^{5 z} - 1\right) \left(- 3 z^{2} - \cos{\left(z \right)} + 1\right)}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
((-1 + exp(5*z))*(1 - cos(z) - 3*z^2) + 5*(z^3 - z + sin(z))*exp(5*z))/(z^3 - z + sin(z))^2
Рациональный знаменатель
[src]
2
/ 3 \ 5*z / 5*z\ / 2\ / 3 \
5*\z - z + sin(z)/ *e + \-1 + e /*\1 - cos(z) - 3*z /*\z - z + sin(z)/
-----------------------------------------------------------------------------
3
/ 3 \
\z - z + sin(z)/
$$\frac{5 \left(z^{3} - z + \sin{\left(z \right)}\right)^{2} e^{5 z} + \left(e^{5 z} - 1\right) \left(- 3 z^{2} - \cos{\left(z \right)} + 1\right) \left(z^{3} - z + \sin{\left(z \right)}\right)}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{3}}$$
5*z 2 5*z 5*z 2 5*z
1 cos(z) e 3*z 5*e cos(z)*e 3*z *e
- ------------------ + ------------------ + ------------------ + ------------------ + --------------- - ------------------ - ------------------
2 2 2 2 3 2 2
/ 3 \ / 3 \ / 3 \ / 3 \ z - z + sin(z) / 3 \ / 3 \
\z - z + sin(z)/ \z - z + sin(z)/ \z - z + sin(z)/ \z - z + sin(z)/ \z - z + sin(z)/ \z - z + sin(z)/
$$- \frac{3 z^{2} e^{5 z}}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}} + \frac{5 e^{5 z}}{z^{3} - z + \sin{\left(z \right)}} - \frac{e^{5 z} \cos{\left(z \right)}}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}} + \frac{e^{5 z}}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}} + \frac{3 z^{2}}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}} + \frac{\cos{\left(z \right)}}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}} - \frac{1}{\left(z^{3} - z + \sin{\left(z \right)}\right)^{2}}$$
-1/(z^3 - z + sin(z))^2 + cos(z)/(z^3 - z + sin(z))^2 + exp(5*z)/(z^3 - z + sin(z))^2 + 3*z^2/(z^3 - z + sin(z))^2 + 5*exp(5*z)/(z^3 - z + sin(z)) - cos(z)*exp(5*z)/(z^3 - z + sin(z))^2 - 3*z^2*exp(5*z)/(z^3 - z + sin(z))^2
Степени
[src]
/ I*z -I*z\
/ 5*z\ | 2 e e |
5*z \-1 + e /*|1 - 3*z - ---- - -----|
5*e \ 2 2 /
--------------------------- + -------------------------------------
/ -I*z I*z\ 2
3 I*\- e + e / / / -I*z I*z\\
z - z - ------------------ | 3 I*\- e + e /|
2 |z - z - ------------------|
\ 2 /
$$\frac{5 e^{5 z}}{z^{3} - z - \frac{i \left(e^{i z} - e^{- i z}\right)}{2}} + \frac{\left(e^{5 z} - 1\right) \left(- 3 z^{2} - \frac{e^{i z}}{2} + 1 - \frac{e^{- i z}}{2}\right)}{\left(z^{3} - z - \frac{i \left(e^{i z} - e^{- i z}\right)}{2}\right)^{2}}$$
5*exp(5*z)/(z^3 - z - i*(-exp(-i*z) + exp(i*z))/2) + (-1 + exp(5*z))*(1 - 3*z^2 - exp(i*z)/2 - exp(-i*z)/2)/(z^3 - z - i*(-exp(-i*z) + exp(i*z))/2)^2
Численный ответ
[src]
5.0*exp(5*z)/(z^3 - z + sin(z)) + (-1.0 + E^(5*z))*(1.0 - cos(z) - 3.0*z^2)/(z^3 - z + sin(z))^2
5.0*exp(5*z)/(z^3 - z + sin(z)) + (-1.0 + E^(5*z))*(1.0 - cos(z) - 3.0*z^2)/(z^3 - z + sin(z))^2
Общий знаменатель
[src]
2 5*z 5*z 2 5*z 3 5*z 5*z 5*z
-1 + 3*z - cos(z)*e - 5*z*e - 3*z *e + 5*z *e + 5*e *sin(z) + cos(z) + e
------------------------------------------------------------------------------------------
2 6 2 4 3
z + z + sin (z) - 2*z - 2*z*sin(z) + 2*z *sin(z)
$$\frac{5 z^{3} e^{5 z} - 3 z^{2} e^{5 z} - 5 z e^{5 z} + 5 e^{5 z} \sin{\left(z \right)} - e^{5 z} \cos{\left(z \right)} + e^{5 z} + 3 z^{2} + \cos{\left(z \right)} - 1}{z^{6} - 2 z^{4} + 2 z^{3} \sin{\left(z \right)} + z^{2} - 2 z \sin{\left(z \right)} + \sin^{2}{\left(z \right)}}$$
(-1 + 3*z^2 - cos(z)*exp(5*z) - 5*z*exp(5*z) - 3*z^2*exp(5*z) + 5*z^3*exp(5*z) + 5*exp(5*z)*sin(z) + cos(z) + exp(5*z))/(z^2 + z^6 + sin(z)^2 - 2*z^4 - 2*z*sin(z) + 2*z^3*sin(z))