Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
((-cos(pi*x/(3+k)))/(2*pi/(3+k))-3/(2*(k-pi))*cos((k-pi)*x/3)+cos(k*x)/k)/3
¿Cómo vas a descomponer esta ((-cos(pi*x/(3+k)))/(2*pi/(3+k))-3/(2*(k-pi))*cos((k-pi)*x/3)+cos(k*x)/k)/3 expresión en fracciones?
Solución
Ha introducido
[src]
/ pi*x\
-cos|-----|
\3 + k/ 3 /(k - pi)*x\ cos(k*x)
------------ - ----------*cos|----------| + --------
/ 2*pi\ 2*(k - pi) \ 3 / k
|-----|
\3 + k/
----------------------------------------------------
3
$$\frac{\left(- \frac{3}{2 \left(k - \pi\right)} \cos{\left(\frac{x \left(k - \pi\right)}{3} \right)} + \frac{\left(-1\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{2 \pi \frac{1}{k + 3}}\right) + \frac{\cos{\left(k x \right)}}{k}}{3}$$
((-cos((pi*x)/(3 + k)))/(((2*pi)/(3 + k))) - 3/((2*(k - pi)))*cos(((k - pi)*x)/3) + cos(k*x)/k)/3
Simplificación general
[src]
/x*(k - pi)\ / pi*x\
- 3*pi*k*cos|----------| + 2*pi*(k - pi)*cos(k*x) - k*(3 + k)*(k - pi)*cos|-----|
\ 3 / \3 + k/
---------------------------------------------------------------------------------
6*pi*k*(k - pi)
$$\frac{- k \left(k + 3\right) \left(k - \pi\right) \cos{\left(\frac{\pi x}{k + 3} \right)} - 3 \pi k \cos{\left(\frac{x \left(k - \pi\right)}{3} \right)} + 2 \pi \left(k - \pi\right) \cos{\left(k x \right)}}{6 \pi k \left(k - \pi\right)}$$
(-3*pi*k*cos(x*(k - pi)/3) + 2*pi*(k - pi)*cos(k*x) - k*(3 + k)*(k - pi)*cos(pi*x/(3 + k)))/(6*pi*k*(k - pi))
Potencias
[src]
/x*(k - pi)\ / pi*x\
cos|----------| (3 + k)*cos|-----|
\ 3 / cos(k*x) \3 + k/
- --------------- + -------- - ------------------
-2*pi + 2*k 3*k 6*pi
$$- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k}$$
/x*(k - pi)\ /3 k\ / pi*x\
cos|----------| |- + -|*cos|-----|
\ 3 / cos(k*x) \2 2/ \3 + k/
- --------------- + -------- - ------------------
-2*pi + 2*k 3*k 3*pi
$$- \frac{\left(\frac{k}{2} + \frac{3}{2}\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{3 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k}$$
-I*x*(k - pi) I*x*(k - pi) / pi*I*x -pi*I*x \
-------------- ------------ | ------ --------|
3 3 I*k*x -I*k*x | 3 + k 3 + k |
e e e e | e e |
--------------- + ------------- ------ + ------- (3 + k)*|- ------- - ---------|
2 2 2 2 \ 2 2 /
- ------------------------------- + ---------------- + -------------------------------
-2*pi + 2*k 3*k 6*pi
$$\frac{\left(k + 3\right) \left(- \frac{e^{\frac{i \pi x}{k + 3}}}{2} - \frac{e^{- \frac{i \pi x}{k + 3}}}{2}\right)}{6 \pi} - \frac{\frac{e^{\frac{i x \left(k - \pi\right)}{3}}}{2} + \frac{e^{- \frac{i x \left(k - \pi\right)}{3}}}{2}}{2 k - 2 \pi} + \frac{\frac{e^{i k x}}{2} + \frac{e^{- i k x}}{2}}{3 k}$$
-(exp(-i*x*(k - pi)/3)/2 + exp(i*x*(k - pi)/3)/2)/(-2*pi + 2*k) + (exp(i*k*x)/2 + exp(-i*k*x)/2)/(3*k) + (3 + k)*(-exp(pi*i*x/(3 + k))/2 - exp(-pi*i*x/(3 + k))/2)/(6*pi)
Unión de expresiones racionales
[src]
/ /x*(k - pi)\ / pi*x\\
k*|- 3*pi*cos|----------| - (3 + k)*(k - pi)*cos|-----|| + 2*pi*(k - pi)*cos(k*x)
\ \ 3 / \3 + k//
---------------------------------------------------------------------------------
6*pi*k*(k - pi)
$$\frac{k \left(- \left(k + 3\right) \left(k - \pi\right) \cos{\left(\frac{\pi x}{k + 3} \right)} - 3 \pi \cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}\right) + 2 \pi \left(k - \pi\right) \cos{\left(k x \right)}}{6 \pi k \left(k - \pi\right)}$$
(k*(-3*pi*cos(x*(k - pi)/3) - (3 + k)*(k - pi)*cos(pi*x/(3 + k))) + 2*pi*(k - pi)*cos(k*x))/(6*pi*k*(k - pi))
Denominador común
[src]
/ pi*x\ 3 / pi*x\ 2 2 / pi*x\ 2 / pi*x\ / pi*x\ / pi*x k*x\
cos|-----| k *cos|-----| + 2*pi *cos(k*x) + 3*k *cos|-----| - k*pi *cos|-----| - 3*pi*k*cos|-----| - 2*pi*k*cos(k*x) + 3*pi*k*cos|- ---- + ---|
\3 + k/ \3 + k/ \3 + k/ \3 + k/ \3 + k/ \ 3 3 /
---------- - ------------------------------------------------------------------------------------------------------------------------------------
6 2 2
- 6*k*pi + 6*pi*k
$$\frac{\cos{\left(\frac{\pi x}{k + 3} \right)}}{6} - \frac{k^{3} \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} - 2 \pi k \cos{\left(k x \right)} - \pi^{2} k \cos{\left(\frac{\pi x}{k + 3} \right)} - 3 \pi k \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 \pi k \cos{\left(\frac{k x}{3} - \frac{\pi x}{3} \right)} + 2 \pi^{2} \cos{\left(k x \right)}}{6 \pi k^{2} - 6 \pi^{2} k}$$
cos(pi*x/(3 + k))/6 - (k^3*cos(pi*x/(3 + k)) + 2*pi^2*cos(k*x) + 3*k^2*cos(pi*x/(3 + k)) - k*pi^2*cos(pi*x/(3 + k)) - 3*pi*k*cos(pi*x/(3 + k)) - 2*pi*k*cos(k*x) + 3*pi*k*cos(-pi*x/3 + k*x/3))/(-6*k*pi^2 + 6*pi*k^2)
Denominador racional
[src]
2 / pi*x\ 2 3 / pi*x\ / pi*x k*x\ 2 / pi*x\ / pi*x\
- 6*k *cos|-----| - 4*pi *cos(k*x) - 2*k *cos|-----| - 6*pi*k*cos|- ---- + ---| + 2*pi*k *cos|-----| + 4*pi*k*cos(k*x) + 6*pi*k*cos|-----|
\3 + k/ \3 + k/ \ 3 3 / \3 + k/ \3 + k/
------------------------------------------------------------------------------------------------------------------------------------------
6*pi*k*(-2*pi + 2*k)
$$\frac{- 2 k^{3} \cos{\left(\frac{\pi x}{k + 3} \right)} - 6 k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} + 2 \pi k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} + 4 \pi k \cos{\left(k x \right)} + 6 \pi k \cos{\left(\frac{\pi x}{k + 3} \right)} - 6 \pi k \cos{\left(\frac{k x}{3} - \frac{\pi x}{3} \right)} - 4 \pi^{2} \cos{\left(k x \right)}}{6 \pi k \left(2 k - 2 \pi\right)}$$
(-6*k^2*cos(pi*x/(3 + k)) - 4*pi^2*cos(k*x) - 2*k^3*cos(pi*x/(3 + k)) - 6*pi*k*cos(-pi*x/3 + k*x/3) + 2*pi*k^2*cos(pi*x/(3 + k)) + 4*pi*k*cos(k*x) + 6*pi*k*cos(pi*x/(3 + k)))/(6*pi*k*(-2*pi + 2*k))
Compilar la expresión
[src]
/(k - pi)*x\ / pi*x\
cos|----------| (3 + k)*cos|-----|
\ 3 / cos(k*x) \3 + k/
- --------------- + -------- - ------------------
-2*pi + 2*k 3*k 6*pi
$$- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k}$$
-cos(((k - pi)*x)/3)/(-2*pi + 2*k) + cos(k*x)/(3*k) - (3 + k)*cos((pi*x)/(3 + k))/(6*pi)
Combinatoria
[src]
/ 3 / pi*x\ 2 2 / pi*x\ 2 / pi*x\ / pi*x\ / pi*x k*x\\
-|k *cos|-----| + 2*pi *cos(k*x) + 3*k *cos|-----| - pi*k *cos|-----| - 3*pi*k*cos|-----| - 2*pi*k*cos(k*x) + 3*pi*k*cos|- ---- + ---||
\ \3 + k/ \3 + k/ \3 + k/ \3 + k/ \ 3 3 //
----------------------------------------------------------------------------------------------------------------------------------------
6*pi*k*(k - pi)
$$- \frac{k^{3} \cos{\left(\frac{\pi x}{k + 3} \right)} - \pi k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 k^{2} \cos{\left(\frac{\pi x}{k + 3} \right)} - 2 \pi k \cos{\left(k x \right)} - 3 \pi k \cos{\left(\frac{\pi x}{k + 3} \right)} + 3 \pi k \cos{\left(\frac{k x}{3} - \frac{\pi x}{3} \right)} + 2 \pi^{2} \cos{\left(k x \right)}}{6 \pi k \left(k - \pi\right)}$$
-(k^3*cos(pi*x/(3 + k)) + 2*pi^2*cos(k*x) + 3*k^2*cos(pi*x/(3 + k)) - pi*k^2*cos(pi*x/(3 + k)) - 3*pi*k*cos(pi*x/(3 + k)) - 2*pi*k*cos(k*x) + 3*pi*k*cos(-pi*x/3 + k*x/3))/(6*pi*k*(k - pi))
Respuesta numérica
[src]
0.333333333333333*cos(k*x)/k - 1.0*cos(((k - pi)*x)/3)/(-6.28318530717959 + 2.0*k) - 0.333333333333333*(0.477464829275686 + 0.159154943091895*k)*cos((pi*x)/(3 + k))
0.333333333333333*cos(k*x)/k - 1.0*cos(((k - pi)*x)/3)/(-6.28318530717959 + 2.0*k) - 0.333333333333333*(0.477464829275686 + 0.159154943091895*k)*cos((pi*x)/(3 + k))
Parte trigonométrica
[src]
/x*(k - pi)\ / pi*x\
cos|----------| (3 + k)*cos|-----|
\ 3 / cos(k*x) \3 + k/
- --------------- + -------- - ------------------
-2*pi + 2*k 3*k 6*pi
$$- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 k - 2 \pi} + \frac{\cos{\left(k x \right)}}{3 k}$$
/pi x*(k - pi)\ /pi \ /pi pi*x\
sin|-- + ----------| sin|-- + k*x| (3 + k)*sin|-- + -----|
\2 3 / \2 / \2 3 + k/
- -------------------- + ------------- - -----------------------
-2*pi + 2*k 3*k 6*pi
$$- \frac{\left(k + 3\right) \sin{\left(\frac{\pi x}{k + 3} + \frac{\pi}{2} \right)}}{6 \pi} - \frac{\sin{\left(\frac{x \left(k - \pi\right)}{3} + \frac{\pi}{2} \right)}}{2 k - 2 \pi} + \frac{\sin{\left(k x + \frac{\pi}{2} \right)}}{3 k}$$
/x*(k - pi)\ / pi*x\
cos|----------| (3 + k)*cos|-----|
\ 3 / cos(k*x) \3 + k/
- --------------- + -------- - ------------------
2*(k - pi) 3*k 6*pi
$$- \frac{\left(k + 3\right) \cos{\left(\frac{\pi x}{k + 3} \right)}}{6 \pi} - \frac{\cos{\left(\frac{x \left(k - \pi\right)}{3} \right)}}{2 \left(k - \pi\right)} + \frac{\cos{\left(k x \right)}}{3 k}$$
2/x*(k - pi)\ 2/k*x\ / 2/ pi*x \\
1 - tan |----------| 1 - tan |---| |1 - tan |---------||*(3 + k)
\ 6 / \ 2 / \ \2*(3 + k)//
- ------------------------------------ + ------------------- - -----------------------------
/ 2/x*(k - pi)\\ / 2/k*x\\ / 2/ pi*x \\
|1 + tan |----------||*(-2*pi + 2*k) 3*k*|1 + tan |---|| 6*pi*|1 + tan |---------||
\ \ 6 // \ \ 2 // \ \2*(3 + k)//
$$- \frac{1 - \tan^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)}}{\left(2 k - 2 \pi\right) \left(\tan^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)} + 1\right)} - \frac{\left(1 - \tan^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)}\right) \left(k + 3\right)}{6 \pi \left(\tan^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{k x}{2} \right)}}{3 k \left(\tan^{2}{\left(\frac{k x}{2} \right)} + 1\right)}$$
1 1 3 + k
- ----------------------------- + ------------ - ---------------
/x*(k - pi)\ 3*k*sec(k*x) / pi*x\
(-2*pi + 2*k)*sec|----------| 6*pi*sec|-----|
\ 3 / \3 + k/
$$- \frac{k + 3}{6 \pi \sec{\left(\frac{\pi x}{k + 3} \right)}} - \frac{1}{\left(2 k - 2 \pi\right) \sec{\left(\frac{x \left(k - \pi\right)}{3} \right)}} + \frac{1}{3 k \sec{\left(k x \right)}}$$
2/x*(k - pi)\ 2/k*x\ / 2/ pi*x \\
-1 + cot |----------| -1 + cot |---| |-1 + cot |---------||*(3 + k)
\ 6 / \ 2 / \ \2*(3 + k)//
- ------------------------------------ + ------------------- - ------------------------------
/ 2/x*(k - pi)\\ / 2/k*x\\ / 2/ pi*x \\
|1 + cot |----------||*(-2*pi + 2*k) 3*k*|1 + cot |---|| 6*pi*|1 + cot |---------||
\ \ 6 // \ \ 2 // \ \2*(3 + k)//
$$- \frac{\left(k + 3\right) \left(\cot^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)} - 1\right)}{6 \pi \left(\cot^{2}{\left(\frac{\pi x}{2 \left(k + 3\right)} \right)} + 1\right)} - \frac{\cot^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)} - 1}{\left(2 k - 2 \pi\right) \left(\cot^{2}{\left(\frac{x \left(k - \pi\right)}{6} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{k x}{2} \right)} - 1}{3 k \left(\cot^{2}{\left(\frac{k x}{2} \right)} + 1\right)}$$
1 1 3 + k
- ---------------------------------- + ----------------- - --------------------
/pi x*(k - pi)\ /pi \ /pi pi*x\
(-2*pi + 2*k)*csc|-- - ----------| 3*k*csc|-- - k*x| 6*pi*csc|-- - -----|
\2 3 / \2 / \2 3 + k/
$$- \frac{k + 3}{6 \pi \csc{\left(- \frac{\pi x}{k + 3} + \frac{\pi}{2} \right)}} - \frac{1}{\left(2 k - 2 \pi\right) \csc{\left(- \frac{x \left(k - \pi\right)}{3} + \frac{\pi}{2} \right)}} + \frac{1}{3 k \csc{\left(- k x + \frac{\pi}{2} \right)}}$$
-1/((-2*pi + 2*k)*csc(pi/2 - x*(k - pi)/3)) + 1/(3*k*csc(pi/2 - k*x)) - (3 + k)/(6*pi*csc(pi/2 - pi*x/(3 + k)))