Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
exp(x)*sin(pi*x)/(1+pi^2)-pi*cos(pi*x)*exp(x)/(1+pi^2)
¿Cómo vas a descomponer esta exp(x)*sin(pi*x)/(1+pi^2)-pi*cos(pi*x)*exp(x)/(1+pi^2) expresión en fracciones?
Solución
Ha introducido
[src]
x x
e *sin(pi*x) pi*cos(pi*x)*e
------------ - ---------------
2 2
1 + pi 1 + pi
$$- \frac{\pi \cos{\left(\pi x \right)} e^{x}}{1 + \pi^{2}} + \frac{e^{x} \sin{\left(\pi x \right)}}{1 + \pi^{2}}$$
(exp(x)*sin(pi*x))/(1 + pi^2) - (pi*cos(pi*x))*exp(x)/(1 + pi^2)
Simplificación general
[src]
x
(-pi*cos(pi*x) + sin(pi*x))*e
------------------------------
2
1 + pi
$$\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}}$$
(-pi*cos(pi*x) + sin(pi*x))*exp(x)/(1 + pi^2)
Respuesta numérica
[src]
0.0919996683503752*exp(x)*sin(pi*x) - 0.289025482222236*cos(pi*x)*exp(x)
0.0919996683503752*exp(x)*sin(pi*x) - 0.289025482222236*cos(pi*x)*exp(x)
Potencias
[src]
/ pi*I*x -pi*I*x\
|e e | x
pi*|------- + --------|*e / -pi*I*x pi*I*x\ x
\ 2 2 / I*\- e + e /*e
- -------------------------- - ---------------------------
2 / 2\
1 + pi 2*\1 + pi /
$$- \frac{\pi \left(\frac{e^{i \pi x}}{2} + \frac{e^{- i \pi x}}{2}\right) e^{x}}{1 + \pi^{2}} - \frac{i \left(e^{i \pi x} - e^{- i \pi x}\right) e^{x}}{2 \left(1 + \pi^{2}\right)}$$
x x
e *sin(pi*x) pi*cos(pi*x)*e
------------ - ---------------
2 2
1 + pi 1 + pi
$$\frac{e^{x} \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}}$$
exp(x)*sin(pi*x)/(1 + pi^2) - pi*cos(pi*x)*exp(x)/(1 + pi^2)
Denominador común
[src]
/ x x\
-\- e *sin(pi*x) + pi*cos(pi*x)*e /
------------------------------------
2
1 + pi
$$- \frac{- e^{x} \sin{\left(\pi x \right)} + \pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}}$$
-(-exp(x)*sin(pi*x) + pi*cos(pi*x)*exp(x))/(1 + pi^2)
Combinatoria
[src]
x
-(-sin(pi*x) + pi*cos(pi*x))*e
--------------------------------
2
1 + pi
$$- \frac{\left(- \sin{\left(\pi x \right)} + \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}}$$
-(-sin(pi*x) + pi*cos(pi*x))*exp(x)/(1 + pi^2)
Denominador racional
[src]
x x
e *sin(pi*x) - pi*cos(pi*x)*e
------------------------------
2
1 + pi
$$\frac{e^{x} \sin{\left(\pi x \right)} - \pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}}$$
(exp(x)*sin(pi*x) - pi*cos(pi*x)*exp(x))/(1 + pi^2)
Compilar la expresión
[src]
x x
e *sin(pi*x) pi*cos(pi*x)*e
------------ - ---------------
2 2
1 + pi 1 + pi
$$\frac{e^{x} \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}}$$
exp(x)*sin(pi*x)/(1 + pi^2) - pi*cos(pi*x)*exp(x)/(1 + pi^2)
Unión de expresiones racionales
[src]
x
(-pi*cos(pi*x) + sin(pi*x))*e
------------------------------
2
1 + pi
$$\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}}$$
(-pi*cos(pi*x) + sin(pi*x))*exp(x)/(1 + pi^2)
Parte trigonométrica
[src]
cosh(x) sinh(x) pi*cosh(x) pi*sinh(x)
---------------- + ---------------- - ---------- - ----------
/ pi \ / pi \ sec(pi*x) sec(pi*x)
sec|- -- + pi*x| sec|- -- + pi*x|
\ 2 / \ 2 /
-------------------------------------------------------------
2
1 + pi
$$\frac{\frac{\sinh{\left(x \right)}}{\sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi \sinh{\left(x \right)}}{\sec{\left(\pi x \right)}} + \frac{\cosh{\left(x \right)}}{\sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi \cosh{\left(x \right)}}{\sec{\left(\pi x \right)}}}{1 + \pi^{2}}$$
/ pi \
(cosh(x) + sinh(x))*cos|- -- + pi*x|
\ 2 / pi*(cosh(x) + sinh(x))*cos(pi*x)
------------------------------------ - --------------------------------
2 2
1 + pi 1 + pi
$$- \frac{\pi \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \cos{\left(\pi x \right)}}{1 + \pi^{2}} + \frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \cos{\left(\pi x - \frac{\pi}{2} \right)}}{1 + \pi^{2}}$$
/ pi \ / pi \
cos|- -- + pi*x|*cosh(x) + cos|- -- + pi*x|*sinh(x) - pi*cos(pi*x)*cosh(x) - pi*cos(pi*x)*sinh(x)
\ 2 / \ 2 /
-------------------------------------------------------------------------------------------------
2
1 + pi
$$\frac{- \pi \cos{\left(\pi x \right)} \sinh{\left(x \right)} - \pi \cos{\left(\pi x \right)} \cosh{\left(x \right)} + \cos{\left(\pi x - \frac{\pi}{2} \right)} \sinh{\left(x \right)} + \cos{\left(\pi x - \frac{\pi}{2} \right)} \cosh{\left(x \right)}}{1 + \pi^{2}}$$
/pi*x\ /pi*x\ / 2/pi*x\\ / 2/pi*x\\
2*cosh(x)*tan|----| 2*sinh(x)*tan|----| pi*|1 - tan |----||*cosh(x) pi*|1 - tan |----||*sinh(x)
\ 2 / \ 2 / \ \ 2 // \ \ 2 //
------------------- + ------------------- - --------------------------- - ---------------------------
2/pi*x\ 2/pi*x\ 2/pi*x\ 2/pi*x\
1 + tan |----| 1 + tan |----| 1 + tan |----| 1 + tan |----|
\ 2 / \ 2 / \ 2 / \ 2 /
-----------------------------------------------------------------------------------------------------
2
1 + pi
$$\frac{- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right) \sinh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} - \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right) \cosh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi x}{2} \right)} \sinh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi x}{2} \right)} \cosh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{1 + \pi^{2}}$$
x
(-pi*cos(pi*x) + sin(pi*x))*e
------------------------------
2
1 + pi
$$\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}}$$
x x
e pi*e
-------------------------- - -------------------
/ 2\ / pi \ / 2\
\1 + pi /*sec|- -- + pi*x| \1 + pi /*sec(pi*x)
\ 2 /
$$\frac{e^{x}}{\left(1 + \pi^{2}\right) \sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi e^{x}}{\left(1 + \pi^{2}\right) \sec{\left(\pi x \right)}}$$
/ 1 pi \ x
|---------------- - ---------|*e
| / pi \ sec(pi*x)|
|sec|- -- + pi*x| |
\ \ 2 / /
---------------------------------
2
1 + pi
$$\frac{\left(\frac{1}{\sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi}{\sec{\left(\pi x \right)}}\right) e^{x}}{1 + \pi^{2}}$$
x x
e pi*e
------------------- - -------------------
/ 2\ / 2\
\1 + pi /*csc(pi*x) \1 + pi /*sec(pi*x)
$$- \frac{\pi e^{x}}{\left(1 + \pi^{2}\right) \sec{\left(\pi x \right)}} + \frac{e^{x}}{\left(1 + \pi^{2}\right) \csc{\left(\pi x \right)}}$$
x x
e pi*e
------------------- - ------------------------
/ 2\ / 2\ /pi \
\1 + pi /*csc(pi*x) \1 + pi /*csc|-- - pi*x|
\2 /
$$- \frac{\pi e^{x}}{\left(1 + \pi^{2}\right) \csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{e^{x}}{\left(1 + \pi^{2}\right) \csc{\left(\pi x \right)}}$$
/pi \
pi*(cosh(x) + sinh(x))*sin|-- + pi*x|
(cosh(x) + sinh(x))*sin(pi*x) \2 /
----------------------------- - -------------------------------------
2 2
1 + pi 1 + pi
$$\frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \sin{\left(\pi x + \frac{\pi}{2} \right)}}{1 + \pi^{2}}$$
cosh(x) sinh(x) pi*cosh(x) pi*sinh(x)
--------- + --------- - -------------- - --------------
csc(pi*x) csc(pi*x) /pi \ /pi \
csc|-- - pi*x| csc|-- - pi*x|
\2 / \2 /
-------------------------------------------------------
2
1 + pi
$$\frac{- \frac{\pi \sinh{\left(x \right)}}{\csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{\sinh{\left(x \right)}}{\csc{\left(\pi x \right)}} - \frac{\pi \cosh{\left(x \right)}}{\csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{\cosh{\left(x \right)}}{\csc{\left(\pi x \right)}}}{1 + \pi^{2}}$$
x /pi*x\ / 2/pi*x\\ x
2*e *tan|----| pi*|1 - tan |----||*e
\ 2 / \ \ 2 //
-------------------------- - --------------------------
/ 2\ / 2/pi*x\\ / 2\ / 2/pi*x\\
\1 + pi /*|1 + tan |----|| \1 + pi /*|1 + tan |----||
\ \ 2 // \ \ 2 //
$$- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right) e^{x}}{\left(1 + \pi^{2}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} + \frac{2 e^{x} \tan{\left(\frac{\pi x}{2} \right)}}{\left(1 + \pi^{2}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}$$
(cosh(x) + sinh(x))*sin(pi*x) pi*(cosh(x) + sinh(x))*cos(pi*x)
----------------------------- - --------------------------------
2 2
1 + pi 1 + pi
$$\frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \cos{\left(\pi x \right)}}{1 + \pi^{2}}$$
/ / pi \\
|-pi*cos(pi*x) + cos|- -- + pi*x||*(cosh(x) + sinh(x))
\ \ 2 //
------------------------------------------------------
2
1 + pi
$$\frac{\left(- \pi \cos{\left(\pi x \right)} + \cos{\left(\pi x - \frac{\pi}{2} \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{1 + \pi^{2}}$$
/ /pi*x\ / 2/pi*x\\\
| 2*cot|----| pi*|-1 + cot |----|||
| \ 2 / \ \ 2 //| x
|-------------- - --------------------|*e
| 2/pi*x\ 2/pi*x\ |
|1 + cot |----| 1 + cot |----| |
\ \ 2 / \ 2 / /
------------------------------------------
2
1 + pi
$$\frac{\left(- \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi x}{2} \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}\right) e^{x}}{1 + \pi^{2}}$$
/ 1 pi \ x
|--------- - --------------|*e
|csc(pi*x) /pi \|
| csc|-- - pi*x||
\ \2 //
-------------------------------
2
1 + pi
$$\frac{\left(- \frac{\pi}{\csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(\pi x \right)}}\right) e^{x}}{1 + \pi^{2}}$$
/pi*x\ /pi*x\ / 2/pi*x\\ / 2/pi*x\\
2*cosh(x)*cot|----| 2*cot|----|*sinh(x) pi*|-1 + cot |----||*cosh(x) pi*|-1 + cot |----||*sinh(x)
\ 2 / \ 2 / \ \ 2 // \ \ 2 //
------------------- + ------------------- - ---------------------------- - ----------------------------
2/pi*x\ 2/pi*x\ 2/pi*x\ 2/pi*x\
1 + cot |----| 1 + cot |----| 1 + cot |----| 1 + cot |----|
\ 2 / \ 2 / \ 2 / \ 2 /
-------------------------------------------------------------------------------------------------------
2
1 + pi
$$\frac{- \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right) \sinh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} - \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right) \cosh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi x}{2} \right)} \sinh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi x}{2} \right)} \cosh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{1 + \pi^{2}}$$
(-pi*cos(pi*x) + sin(pi*x))*(cosh(x) + sinh(x))
-----------------------------------------------
2
1 + pi
$$\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{1 + \pi^{2}}$$
/pi \ /pi \
cosh(x)*sin(pi*x) + sin(pi*x)*sinh(x) - pi*cosh(x)*sin|-- + pi*x| - pi*sin|-- + pi*x|*sinh(x)
\2 / \2 /
---------------------------------------------------------------------------------------------
2
1 + pi
$$\frac{\sin{\left(\pi x \right)} \sinh{\left(x \right)} + \sin{\left(\pi x \right)} \cosh{\left(x \right)} - \pi \sin{\left(\pi x + \frac{\pi}{2} \right)} \sinh{\left(x \right)} - \pi \sin{\left(\pi x + \frac{\pi}{2} \right)} \cosh{\left(x \right)}}{1 + \pi^{2}}$$
/ /pi*x\ / 2/pi*x\\\
| 2*tan|----| pi*|1 - tan |----|||
| \ 2 / \ \ 2 //| x
|-------------- - -------------------|*e
| 2/pi*x\ 2/pi*x\ |
|1 + tan |----| 1 + tan |----| |
\ \ 2 / \ 2 / /
-----------------------------------------
2
1 + pi
$$\frac{\left(- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi x}{2} \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}\right) e^{x}}{1 + \pi^{2}}$$
cosh(x)*sin(pi*x) + sin(pi*x)*sinh(x) - pi*cos(pi*x)*cosh(x) - pi*cos(pi*x)*sinh(x)
-----------------------------------------------------------------------------------
2
1 + pi
$$\frac{\sin{\left(\pi x \right)} \sinh{\left(x \right)} + \sin{\left(\pi x \right)} \cosh{\left(x \right)} - \pi \cos{\left(\pi x \right)} \sinh{\left(x \right)} - \pi \cos{\left(\pi x \right)} \cosh{\left(x \right)}}{1 + \pi^{2}}$$
/ /pi \ \
|- pi*sin|-- + pi*x| + sin(pi*x)|*(cosh(x) + sinh(x))
\ \2 / /
-----------------------------------------------------
2
1 + pi
$$\frac{\left(\sin{\left(\pi x \right)} - \pi \sin{\left(\pi x + \frac{\pi}{2} \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{1 + \pi^{2}}$$
/pi*x\ x / 2/pi*x\\ x
2*cot|----|*e pi*|-1 + cot |----||*e
\ 2 / \ \ 2 //
-------------------------- - --------------------------
/ 2\ / 2/pi*x\\ / 2\ / 2/pi*x\\
\1 + pi /*|1 + cot |----|| \1 + pi /*|1 + cot |----||
\ \ 2 // \ \ 2 //
$$- \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right) e^{x}}{\left(1 + \pi^{2}\right) \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} + \frac{2 e^{x} \cot{\left(\frac{\pi x}{2} \right)}}{\left(1 + \pi^{2}\right) \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}$$
2*cot(pi*x/2)*exp(x)/((1 + pi^2)*(1 + cot(pi*x/2)^2)) - pi*(-1 + cot(pi*x/2)^2)*exp(x)/((1 + pi^2)*(1 + cot(pi*x/2)^2))