Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
exp^(t)*sqrt(2)*cos(t+(pi/4))/(cos(t+(pi/4)))^2
¿Cómo vas a descomponer esta exp^(t)*sqrt(2)*cos(t+(pi/4))/(cos(t+(pi/4)))^2 expresión en fracciones?
Solución
Ha introducido
[src]
t ___ / pi\
E *\/ 2 *cos|t + --|
\ 4 /
--------------------
2/ pi\
cos |t + --|
\ 4 /
$$\frac{\sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}}{\cos^{2}{\left(t + \frac{\pi}{4} \right)}}$$
((E^t*sqrt(2))*cos(t + pi/4))/cos(t + pi/4)^2
Simplificación general
[src]
___ t \/ 2 *e ----------- / pi\ cos|t + --| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos(t + pi/4)
Respuesta numérica
[src]
1.4142135623731*2.71828182845905^t/cos(t + pi/4)
1.4142135623731*2.71828182845905^t/cos(t + pi/4)
Combinatoria
[src]
___ t \/ 2 *e ----------- / pi\ cos|t + --| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos(t + pi/4)
Unión de expresiones racionales
[src]
___ t \/ 2 *e ------------- /pi + 4*t\ cos|--------| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(\frac{4 t + \pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos((pi + 4*t)/4)
Denominador común
[src]
___ t \/ 2 *e ----------- / pi\ cos|t + --| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos(t + pi/4)
Potencias
[src]
___ t
\/ 2 *e
--------------------------
/ pi\ / pi\
I*|t + --| I*|-t - --|
\ 4 / \ 4 /
e e
----------- + ------------
2 2
$$\frac{\sqrt{2} e^{t}}{\frac{e^{i \left(- t - \frac{\pi}{4}\right)}}{2} + \frac{e^{i \left(t + \frac{\pi}{4}\right)}}{2}}$$
___ t \/ 2 *e ----------- / pi\ cos|t + --| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos(t + pi/4)
Compilar la expresión
[src]
___ t \/ 2 *e ----------- / pi\ cos|t + --| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos(t + pi/4)
Parte trigonométrica
[src]
___ t / 2/t pi\\
\/ 2 *(cosh(1) + sinh(1)) *|1 + tan |- + --||
\ \2 8 //
---------------------------------------------
2/t pi\
1 - tan |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t}}{1 - \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$
___
\/ 2 *(cosh(t) + sinh(t))
----------------------------------
/ 2/ pi\ \
| sin |t + --| |
| \ 4 / | 2/t pi\
|-1 + --------------|*sin |- + --|
| 4/t pi\| \2 8 /
| 4*sin |- + --||
\ \2 8 //
$$\frac{\sqrt{2} \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\left(-1 + \frac{\sin^{2}{\left(t + \frac{\pi}{4} \right)}}{4 \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}\right) \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$
___ / 2/t pi\\ t
\/ 2 *|1 + tan |- + --||*e
\ \2 8 //
---------------------------
2/t pi\
1 - tan |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) e^{t}}{1 - \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$
/ 4/t pi\\
| 4*sin |- + --||
___ | \2 8 /|
\/ 2 *|1 + --------------|*(cosh(t) + sinh(t))
| 2/ pi\ |
| sin |t + --| |
\ \ 4 / /
----------------------------------------------
4/t pi\
4*sin |- + --|
\2 8 /
1 - --------------
2/ pi\
sin |t + --|
\ 4 /
$$\frac{\sqrt{2} \left(\frac{4 \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\sin^{2}{\left(t + \frac{\pi}{4} \right)}} + 1\right) \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{- \frac{4 \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\sin^{2}{\left(t + \frac{\pi}{4} \right)}} + 1}$$
/ 2/t pi\ \
| sec |- + --| |
___ | \2 8 / | t
\/ 2 *|1 + --------------|*e
| 2/t 3*pi\|
| sec |- - ----||
\ \2 8 //
-----------------------------
2/t pi\
sec |- + --|
\2 8 /
1 - --------------
2/t 3*pi\
sec |- - ----|
\2 8 /
$$\frac{\sqrt{2} \left(1 + \frac{\sec^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}\right) e^{t}}{1 - \frac{\sec^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}}$$
___ t / 2/t pi\\
-\/ 2 *(cosh(1) + sinh(1)) *|1 + tan |- + --||
\ \2 8 //
-----------------------------------------------
2/t pi\
-1 + tan |- + --|
\2 8 /
$$- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1}$$
___ t / pi\
\/ 2 *e *sec|t + --|
\ 4 /
$$\sqrt{2} e^{t} \sec{\left(t + \frac{\pi}{4} \right)}$$
___ / 2/t pi\\
-\/ 2 *|1 + tan |- + --||*(cosh(t) + sinh(t))
\ \2 8 //
----------------------------------------------
2/t pi\
-1 + tan |- + --|
\2 8 /
$$- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1}$$
___ 2/t 3*pi\ t
\/ 2 *sec |- - ----|*e
\2 8 /
-----------------------
1
-1 + ------------
2/t pi\
tan |- + --|
\2 8 /
$$\frac{\sqrt{2} e^{t} \sec^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}{-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}}$$
___ t / 2/t pi\\
\/ 2 *(cosh(1) + sinh(1)) *|1 + cot |- + --||
\ \2 8 //
---------------------------------------------
2/t pi\
-1 + cot |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t}}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1}$$
___ t
\/ 2 *e
--------------------------------
/ 2/t pi\\ 2/t pi\
|-1 + cot |- + --||*sin |- + --|
\ \2 8 // \2 8 /
$$\frac{\sqrt{2} e^{t}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$
___ / pi\ t
2*\/ 2 *cos|t + --|*e
\ 4 /
----------------------
1 - sin(2*t)
$$\frac{2 \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}}{1 - \sin{\left(2 t \right)}}$$
___
\/ 2 *(cosh(t) + sinh(t))
-------------------------
/ 3*pi\
sin|t + ----|
\ 4 /
$$\frac{\sqrt{2} \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\sin{\left(t + \frac{3 \pi}{4} \right)}}$$
___ t \/ 2 *(cosh(1) + sinh(1)) -------------------------------- / 2/t pi\\ 2/t pi\ |-1 + cot |- + --||*sin |- + --| \ \2 8 // \2 8 /
$$\frac{\sqrt{2} \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t}}{\left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$
/ 2/t 3*pi\\
| cos |- - ----||
___ | \2 8 /|
\/ 2 *|1 + --------------|*(cosh(t) + sinh(t))
| 2/t pi\ |
| cos |- + --| |
\ \2 8 / /
----------------------------------------------
2/t 3*pi\
cos |- - ----|
\2 8 /
1 - --------------
2/t pi\
cos |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(\frac{\cos^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}{\cos^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}} + 1\right) \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{- \frac{\cos^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}{\cos^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}} + 1}$$
___ / 2/t pi\\ t
\/ 2 *|1 + cot |- + --||*e
\ \2 8 //
---------------------------
2/t pi\
-1 + cot |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} + 1\right) e^{t}}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1}$$
___
\/ 2 *(cosh(t) + sinh(t))
-------------------------
/ pi\
cos|t + --|
\ 4 /
$$\frac{\sqrt{2} \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
___
\/ 2 *(cosh(t) + sinh(t))
------------------------------------
/ 2/t pi\ \
| cos |- + --| |
| \2 8 / | 2/t 3*pi\
|-1 + --------------|*cos |- - ----|
| 2/t 3*pi\| \2 8 /
| cos |- - ----||
\ \2 8 //
$$\frac{\sqrt{2} \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\left(-1 + \frac{\cos^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}\right) \cos^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}$$
2
___ / 2/t pi\\ t
\/ 2 *|1 + cot |- + --|| *e
\ \4 16//
----------------------------------
/ 2/t pi\\ 2/t pi\
4*|-1 + cot |- + --||*cot |- + --|
\ \2 8 // \4 16/
$$\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{t}{4} + \frac{\pi}{16} \right)} + 1\right)^{2} e^{t}}{4 \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1\right) \cot^{2}{\left(\frac{t}{4} + \frac{\pi}{16} \right)}}$$
___ 2/t pi\ t
\/ 2 *csc |- + --|*e
\2 8 /
---------------------
1
-1 + ------------
2/t pi\
tan |- + --|
\2 8 /
$$\frac{\sqrt{2} e^{t} \csc^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}}$$
___ 2/t pi\ t
\/ 2 *tan |- + --|*e
\4 16/
----------------------------------
/ 2/t pi\\ 4/t pi\
4*|-1 + cot |- + --||*sin |- + --|
\ \2 8 // \4 16/
$$\frac{\sqrt{2} e^{t} \tan^{2}{\left(\frac{t}{4} + \frac{\pi}{16} \right)}}{4 \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{4}{\left(\frac{t}{4} + \frac{\pi}{16} \right)}}$$
___ t
\/ 2 *(cosh(1) + sinh(1))
--------------------------
/ 3*pi\
sin|t + ----|
\ 4 /
$$\frac{\sqrt{2} \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t}}{\sin{\left(t + \frac{3 \pi}{4} \right)}}$$
___ t / pi\
\/ 2 *(cosh(1) + sinh(1)) *csc|-t + --|
\ 4 /
$$\sqrt{2} \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t} \csc{\left(- t + \frac{\pi}{4} \right)}$$
/ 2/ t 3*pi\\
| csc |- - + ----||
___ | \ 2 8 /| t
\/ 2 *|1 + ----------------|*e
| 2/t pi\ |
| csc |- + --| |
\ \2 8 / /
-------------------------------
2/ t 3*pi\
csc |- - + ----|
\ 2 8 /
1 - ----------------
2/t pi\
csc |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(\frac{\csc^{2}{\left(- \frac{t}{2} + \frac{3 \pi}{8} \right)}}{\csc^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}} + 1\right) e^{t}}{- \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{3 \pi}{8} \right)}}{\csc^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}} + 1}$$
___ t / pi\
\/ 2 *(cosh(1) + sinh(1)) *sec|t + --|
\ 4 /
$$\sqrt{2} \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t} \sec{\left(t + \frac{\pi}{4} \right)}$$
___ -\/ 2 *(cosh(t) + sinh(t)) --------------------------------- / 2/ pi\ \ | sin |t + --| | | \ 4 / | 2/t pi\ |1 - --------------|*sin |- + --| | 4/t pi\| \2 8 / | 4*sin |- + --|| \ \2 8 //
$$- \frac{\sqrt{2} \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\left(1 - \frac{\sin^{2}{\left(t + \frac{\pi}{4} \right)}}{4 \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}\right) \sin^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}$$
/ 4/t pi\\
| 8*sin |- + --||
___ | \2 8 /|
\/ 2 *|1 + --------------|*(cosh(t) + sinh(t))
\ 1 + sin(2*t) /
----------------------------------------------
4/t pi\
8*sin |- + --|
\2 8 /
1 - --------------
1 + sin(2*t)
$$\frac{\sqrt{2} \left(1 + \frac{8 \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\sin{\left(2 t \right)} + 1}\right) \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{1 - \frac{8 \sin^{4}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\sin{\left(2 t \right)} + 1}}$$
___ 2/t 3*pi\ t
\/ 2 *sec |- - ----|*e
\2 8 /
-----------------------
2/t 3*pi\
sec |- - ----|
\2 8 /
-1 + --------------
2/t pi\
sec |- + --|
\2 8 /
$$\frac{\sqrt{2} e^{t} \sec^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}{\frac{\sec^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}{\sec^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}} - 1}$$
___ t
\/ 2 *(cosh(1) + sinh(1))
--------------------------
/ pi\
cos|t + --|
\ 4 /
$$\frac{\sqrt{2} \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
___ / 2/t 3*pi\\ t
\/ 2 *|1 + tan |- + ----||*e
\ \2 8 //
-----------------------------
/t 3*pi\
2*tan|- + ----|
\2 8 /
$$\frac{\sqrt{2} \left(\tan^{2}{\left(\frac{t}{2} + \frac{3 \pi}{8} \right)} + 1\right) e^{t}}{2 \tan{\left(\frac{t}{2} + \frac{3 \pi}{8} \right)}}$$
___ / 1 \ t
\/ 2 *|1 + ------------|*e
| 2/t pi\|
| cot |- + --||
\ \2 8 //
---------------------------
1
1 - ------------
2/t pi\
cot |- + --|
\2 8 /
$$\frac{\sqrt{2} \left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}\right) e^{t}}{1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}}$$
___ 2/t pi\ t
\/ 2 *csc |- + --|*e
\2 8 /
---------------------
2/t pi\
csc |- + --|
\2 8 /
-1 + ----------------
2/ t 3*pi\
csc |- - + ----|
\ 2 8 /
$$\frac{\sqrt{2} e^{t} \csc^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{-1 + \frac{\csc^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{3 \pi}{8} \right)}}}$$
___ -\/ 2 *(cosh(t) + sinh(t)) --------------------------------- / 1 \ 2/t 3*pi\ |1 - ------------|*cos |- - ----| | 2/t pi\| \2 8 / | tan |- + --|| \ \2 8 //
$$- \frac{\sqrt{2} \left(\sinh{\left(t \right)} + \cosh{\left(t \right)}\right)}{\left(1 - \frac{1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}\right) \cos^{2}{\left(\frac{t}{2} - \frac{3 \pi}{8} \right)}}$$
2
___ / 2/t pi\\ t
\/ 2 *|1 + tan |- + --|| *e
\ \4 16//
----------------------------------
/ 1 \ 2/t pi\
4*|-1 + ------------|*tan |- + --|
| 2/t pi\| \4 16/
| tan |- + --||
\ \2 8 //
$$\frac{\sqrt{2} \left(\tan^{2}{\left(\frac{t}{4} + \frac{\pi}{16} \right)} + 1\right)^{2} e^{t}}{4 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{8} \right)}}\right) \tan^{2}{\left(\frac{t}{4} + \frac{\pi}{16} \right)}}$$
___ / pi\ t
\/ 2 *csc|-t + --|*e
\ 4 /
$$\sqrt{2} e^{t} \csc{\left(- t + \frac{\pi}{4} \right)}$$
sqrt(2)*csc(-t + pi/4)*exp(t)
Denominador racional
[src]
___ t \/ 2 *e ----------- / pi\ cos|t + --| \ 4 /
$$\frac{\sqrt{2} e^{t}}{\cos{\left(t + \frac{\pi}{4} \right)}}$$
sqrt(2)*exp(t)/cos(t + pi/4)