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