Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(u*x)*((cos(u*pi/2))/(pi*(1-u^2)))+sin(u*x)*((sin(u*pi/2)-u)/(pi*(1-u^2)))
¿Cómo vas a descomponer esta cos(u*x)*((cos(u*pi/2))/(pi*(1-u^2)))+sin(u*x)*((sin(u*pi/2)-u)/(pi*(1-u^2))) expresión en fracciones?
Solución
Ha introducido
[src]
/u*pi\ /u*pi\
cos|----| sin|----| - u
\ 2 / \ 2 /
cos(u*x)*----------- + sin(u*x)*-------------
/ 2\ / 2\
pi*\1 - u / pi*\1 - u /
$$\frac{- u + \sin{\left(\frac{\pi u}{2} \right)}}{\pi \left(1 - u^{2}\right)} \sin{\left(u x \right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)}}{\pi \left(1 - u^{2}\right)} \cos{\left(u x \right)}$$
cos(u*x)*(cos((u*pi)/2)/((pi*(1 - u^2)))) + sin(u*x)*((sin((u*pi)/2) - u)/((pi*(1 - u^2))))
Simplificación general
[src]
/ / pi\\
- cos|u*|x - --|| + u*sin(u*x)
\ \ 2 //
------------------------------
/ 2\
pi*\-1 + u /
$$\frac{u \sin{\left(u x \right)} - \cos{\left(u \left(x - \frac{\pi}{2}\right) \right)}}{\pi \left(u^{2} - 1\right)}$$
(-cos(u*(x - pi/2)) + u*sin(u*x))/(pi*(-1 + u^2))
Respuesta numérica
[src]
(-u + sin((u*pi)/2))*sin(u*x)/(3.14159265358979 - 3.14159265358979*u^2) + cos(u*x)*cos((u*pi)/2)/(3.14159265358979 - 3.14159265358979*u^2)
(-u + sin((u*pi)/2))*sin(u*x)/(3.14159265358979 - 3.14159265358979*u^2) + cos(u*x)*cos((u*pi)/2)/(3.14159265358979 - 3.14159265358979*u^2)
Unión de expresiones racionales
[src]
/ /pi*u\\ /pi*u\
|-u + sin|----||*sin(u*x) + cos(u*x)*cos|----|
\ \ 2 // \ 2 /
----------------------------------------------
/ 2\
pi*\1 - u /
$$\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)} + \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)}$$
((-u + sin(pi*u/2))*sin(u*x) + cos(u*x)*cos(pi*u/2))/(pi*(1 - u^2))
Combinatoria
[src]
/pi*u\ /pi*u\
u*sin(u*x) - cos(u*x)*cos|----| - sin(u*x)*sin|----|
\ 2 / \ 2 /
----------------------------------------------------
pi*(1 + u)*(-1 + u)
$$\frac{u \sin{\left(u x \right)} - \sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)} - \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(u - 1\right) \left(u + 1\right)}$$
(u*sin(u*x) - cos(u*x)*cos(pi*u/2) - sin(u*x)*sin(pi*u/2))/(pi*(1 + u)*(-1 + u))
Potencias
[src]
/ pi*I*u -pi*I*u \ / / -pi*I*u pi*I*u\\
| ------ --------| | | -------- ------||
/ I*u*x -I*u*x\ | 2 2 | | | 2 2 ||
|e e | |e e | | I*\- e + e /| / -I*u*x I*u*x\
|------ + -------|*|------- + ---------| I*|-u - -------------------------|*\- e + e /
\ 2 2 / \ 2 2 / \ 2 /
---------------------------------------- - -------------------------------------------------------
/ 2\ / 2\
pi*\1 - u / 2*pi*\1 - u /
$$- \frac{i \left(- u - \frac{i \left(e^{\frac{i \pi u}{2}} - e^{- \frac{i \pi u}{2}}\right)}{2}\right) \left(e^{i u x} - e^{- i u x}\right)}{2 \pi \left(1 - u^{2}\right)} + \frac{\left(\frac{e^{\frac{i \pi u}{2}}}{2} + \frac{e^{- \frac{i \pi u}{2}}}{2}\right) \left(\frac{e^{i u x}}{2} + \frac{e^{- i u x}}{2}\right)}{\pi \left(1 - u^{2}\right)}$$
/ /pi*u\\ /pi*u\
|-u + sin|----||*sin(u*x) cos(u*x)*cos|----|
\ \ 2 // \ 2 /
------------------------- + ------------------
/ 2\ / 2\
pi*\1 - u / pi*\1 - u /
$$\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)}$$
(-u + sin(pi*u/2))*sin(u*x)/(pi*(1 - u^2)) + cos(u*x)*cos(pi*u/2)/(pi*(1 - u^2))
Denominador racional
[src]
/pi*u\ /pi*u\
u*sin(u*x) - cos(u*x)*cos|----| - sin(u*x)*sin|----|
\ 2 / \ 2 /
----------------------------------------------------
/ 2\
pi*\-1 + u /
$$\frac{u \sin{\left(u x \right)} - \sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)} - \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(u^{2} - 1\right)}$$
(u*sin(u*x) - cos(u*x)*cos(pi*u/2) - sin(u*x)*sin(pi*u/2))/(pi*(-1 + u^2))
Compilar la expresión
[src]
/ /u*pi\\ /u*pi\
|-u + sin|----||*sin(u*x) cos(u*x)*cos|----|
\ \ 2 // \ 2 /
------------------------- + ------------------
/ 2\ / 2\
pi*\1 - u / pi*\1 - u /
$$\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)}$$
(-u + sin((u*pi)/2))*sin(u*x)/(pi*(1 - u^2)) + cos(u*x)*cos((u*pi)/2)/(pi*(1 - u^2))
Denominador común
[src]
/pi*u\ /pi*u\
u*sin(u*x) - cos(u*x)*cos|----| - sin(u*x)*sin|----|
\ 2 / \ 2 /
----------------------------------------------------
2
-pi + pi*u
$$\frac{u \sin{\left(u x \right)} - \sin{\left(\frac{\pi u}{2} \right)} \sin{\left(u x \right)} - \cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi u^{2} - \pi}$$
(u*sin(u*x) - cos(u*x)*cos(pi*u/2) - sin(u*x)*sin(pi*u/2))/(-pi + pi*u^2)
Parte trigonométrica
[src]
/ / pi\\
|u*|x - --||
2| \ 2 /| /u*x\
1 - tan |----------| 2*u*tan|---|
\ 2 / \ 2 /
- -------------------- + -------------
/ / pi\\ 2/u*x\
|u*|x - --|| 1 + tan |---|
2| \ 2 /| \ 2 /
1 + tan |----------|
\ 2 /
--------------------------------------
/ 2\
pi*\-1 + u /
$$\frac{\frac{2 u \tan{\left(\frac{u x}{2} \right)}}{\tan^{2}{\left(\frac{u x}{2} \right)} + 1} - \frac{1 - \tan^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)}}{\tan^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)} + 1}}{\pi \left(u^{2} - 1\right)}$$
/ /pi*u\\ /pi*u\
|-u + sin|----||*sin(u*x) cos(u*x)*cos|----|
\ \ 2 // \ 2 /
------------------------- + ------------------
/ 2\ / 2\
pi*\1 - u / pi*\1 - u /
$$\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)}$$
/ / pi\\
|u*|x - --||
2| \ 2 /| /u*x\
-1 + cot |----------| 2*u*cot|---|
\ 2 / \ 2 /
- --------------------- + -------------
/ / pi\\ 2/u*x\
|u*|x - --|| 1 + cot |---|
2| \ 2 /| \ 2 /
1 + cot |----------|
\ 2 /
---------------------------------------
/ 2\
pi*\-1 + u /
$$\frac{\frac{2 u \cot{\left(\frac{u x}{2} \right)}}{\cot^{2}{\left(\frac{u x}{2} \right)} + 1} - \frac{\cot^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)} - 1}{\cot^{2}{\left(\frac{u \left(x - \frac{\pi}{2}\right)}{2} \right)} + 1}}{\pi \left(u^{2} - 1\right)}$$
/pi / pi\\
- sin|-- + u*|x - --|| + u*sin(u*x)
\2 \ 2 //
-----------------------------------
/ 2\
pi*\-1 + u /
$$\frac{u \sin{\left(u x \right)} - \sin{\left(u \left(x - \frac{\pi}{2}\right) + \frac{\pi}{2} \right)}}{\pi \left(u^{2} - 1\right)}$$
1 u
- --------------- + ---------------
/ / pi\\ / pi \
sec|u*|x - --|| sec|- -- + u*x|
\ \ 2 // \ 2 /
-----------------------------------
/ 2\
pi*\-1 + u /
$$\frac{\frac{u}{\sec{\left(u x - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(u \left(x - \frac{\pi}{2}\right) \right)}}}{\pi \left(u^{2} - 1\right)}$$
/ / pi pi*u\\ / pi \ /pi*u\
|-u + cos|- -- + ----||*cos|- -- + u*x| cos(u*x)*cos|----|
\ \ 2 2 // \ 2 / \ 2 /
--------------------------------------- + ------------------
/ 2\ / 2\
pi*\1 - u / pi*\1 - u /
$$\frac{\left(- u + \cos{\left(\frac{\pi u}{2} - \frac{\pi}{2} \right)}\right) \cos{\left(u x - \frac{\pi}{2} \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\cos{\left(\frac{\pi u}{2} \right)} \cos{\left(u x \right)}}{\pi \left(1 - u^{2}\right)}$$
1 u
- -------------------- + --------
/pi / pi\\ csc(u*x)
csc|-- - u*|x - --||
\2 \ 2 //
---------------------------------
/ 2\
pi*\-1 + u /
$$\frac{\frac{u}{\csc{\left(u x \right)}} - \frac{1}{\csc{\left(- u \left(x - \frac{\pi}{2}\right) + \frac{\pi}{2} \right)}}}{\pi \left(u^{2} - 1\right)}$$
1
---------------- - u
/ pi pi*u\
sec|- -- + ----|
1 \ 2 2 /
------------------------------ + ---------------------------
/ 2\ /pi*u\ / 2\ / pi \
pi*\1 - u /*sec(u*x)*sec|----| pi*\1 - u /*sec|- -- + u*x|
\ 2 / \ 2 /
$$\frac{- u + \frac{1}{\sec{\left(\frac{\pi u}{2} - \frac{\pi}{2} \right)}}}{\pi \left(1 - u^{2}\right) \sec{\left(u x - \frac{\pi}{2} \right)}} + \frac{1}{\pi \left(1 - u^{2}\right) \sec{\left(\frac{\pi u}{2} \right)} \sec{\left(u x \right)}}$$
/ / pi\\
- cos|u*|x - --|| + u*sin(u*x)
\ \ 2 //
------------------------------
/ 2\
pi*\-1 + u /
$$\frac{u \sin{\left(u x \right)} - \cos{\left(u \left(x - \frac{\pi}{2}\right) \right)}}{\pi \left(u^{2} - 1\right)}$$
/ /pi*u\ \
| 2*tan|----| |
| \ 4 / | /u*x\
2*|-u + --------------|*tan|---|
| 2/pi*u\| \ 2 / / 2/u*x\\ / 2/pi*u\\
| 1 + tan |----|| |1 - tan |---||*|1 - tan |----||
\ \ 4 // \ \ 2 // \ \ 4 //
-------------------------------- + --------------------------------------------
/ 2/u*x\\ / 2\ / 2/u*x\\ / 2/pi*u\\ / 2\
pi*|1 + tan |---||*\1 - u / pi*|1 + tan |---||*|1 + tan |----||*\1 - u /
\ \ 2 // \ \ 2 // \ \ 4 //
$$\frac{\left(1 - \tan^{2}{\left(\frac{\pi u}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{u x}{2} \right)}\right)}{\pi \left(1 - u^{2}\right) \left(\tan^{2}{\left(\frac{\pi u}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{u x}{2} \right)} + 1\right)} + \frac{2 \left(- u + \frac{2 \tan{\left(\frac{\pi u}{4} \right)}}{\tan^{2}{\left(\frac{\pi u}{4} \right)} + 1}\right) \tan{\left(\frac{u x}{2} \right)}}{\pi \left(1 - u^{2}\right) \left(\tan^{2}{\left(\frac{u x}{2} \right)} + 1\right)}$$
1
--------- - u
/pi*u\
csc|----|
\ 2 / 1
-------------------- + ------------------------------
/ 2\ / 2\ /pi*u\
pi*\1 - u /*csc(u*x) pi*\1 - u /*sec(u*x)*sec|----|
\ 2 /
$$\frac{- u + \frac{1}{\csc{\left(\frac{\pi u}{2} \right)}}}{\pi \left(1 - u^{2}\right) \csc{\left(u x \right)}} + \frac{1}{\pi \left(1 - u^{2}\right) \sec{\left(\frac{\pi u}{2} \right)} \sec{\left(u x \right)}}$$
/ /pi*u\\ /pi \ /pi pi*u\
|-u + sin|----||*sin(u*x) sin|-- + u*x|*sin|-- + ----|
\ \ 2 // \2 / \2 2 /
------------------------- + ----------------------------
/ 2\ / 2\
pi*\1 - u / pi*\1 - u /
$$\frac{\left(- u + \sin{\left(\frac{\pi u}{2} \right)}\right) \sin{\left(u x \right)}}{\pi \left(1 - u^{2}\right)} + \frac{\sin{\left(\frac{\pi u}{2} + \frac{\pi}{2} \right)} \sin{\left(u x + \frac{\pi}{2} \right)}}{\pi \left(1 - u^{2}\right)}$$
/ / pi\\ / pi \
- cos|u*|x - --|| + u*cos|- -- + u*x|
\ \ 2 // \ 2 /
-------------------------------------
/ 2\
pi*\-1 + u /
$$\frac{u \cos{\left(u x - \frac{\pi}{2} \right)} - \cos{\left(u \left(x - \frac{\pi}{2}\right) \right)}}{\pi \left(u^{2} - 1\right)}$$
1
--------- - u
/pi*u\
csc|----|
\ 2 / 1
-------------------- + ----------------------------------------
/ 2\ / 2\ /pi \ /pi pi*u\
pi*\1 - u /*csc(u*x) pi*\1 - u /*csc|-- - u*x|*csc|-- - ----|
\2 / \2 2 /
$$\frac{- u + \frac{1}{\csc{\left(\frac{\pi u}{2} \right)}}}{\pi \left(1 - u^{2}\right) \csc{\left(u x \right)}} + \frac{1}{\pi \left(1 - u^{2}\right) \csc{\left(- \frac{\pi u}{2} + \frac{\pi}{2} \right)} \csc{\left(- u x + \frac{\pi}{2} \right)}}$$
/ /pi*u\ \
| 2*cot|----| |
| \ 4 / | /u*x\
2*|-u + --------------|*cot|---|
| 2/pi*u\| \ 2 / / 2/u*x\\ / 2/pi*u\\
| 1 + cot |----|| |-1 + cot |---||*|-1 + cot |----||
\ \ 4 // \ \ 2 // \ \ 4 //
-------------------------------- + --------------------------------------------
/ 2/u*x\\ / 2\ / 2/u*x\\ / 2/pi*u\\ / 2\
pi*|1 + cot |---||*\1 - u / pi*|1 + cot |---||*|1 + cot |----||*\1 - u /
\ \ 2 // \ \ 2 // \ \ 4 //
$$\frac{2 \left(- u + \frac{2 \cot{\left(\frac{\pi u}{4} \right)}}{\cot^{2}{\left(\frac{\pi u}{4} \right)} + 1}\right) \cot{\left(\frac{u x}{2} \right)}}{\pi \left(1 - u^{2}\right) \left(\cot^{2}{\left(\frac{u x}{2} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(\frac{\pi u}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{u x}{2} \right)} - 1\right)}{\pi \left(1 - u^{2}\right) \left(\cot^{2}{\left(\frac{\pi u}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{u x}{2} \right)} + 1\right)}$$
2*(-u + 2*cot(pi*u/4)/(1 + cot(pi*u/4)^2))*cot(u*x/2)/(pi*(1 + cot(u*x/2)^2)*(1 - u^2)) + (-1 + cot(u*x/2)^2)*(-1 + cot(pi*u/4)^2)/(pi*(1 + cot(u*x/2)^2)*(1 + cot(pi*u/4)^2)*(1 - u^2))