Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(pi/((6*(x^2+1))))/5-pi*x^2*cos(pi/(6*(x^2+1)))/(15*(x^2+1)^2)
¿Cómo vas a descomponer esta sin(pi/((6*(x^2+1))))/5-pi*x^2*cos(pi/(6*(x^2+1)))/(15*(x^2+1)^2) expresión en fracciones?
Solución
Ha introducido
[src]
/ pi \ 2 / pi \
sin|----------| pi*x *cos|----------|
| / 2 \| | / 2 \|
\6*\x + 1// \6*\x + 1//
--------------- - ---------------------
5 2
/ 2 \
15*\x + 1/
$$- \frac{\pi x^{2} \cos{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{5}$$
sin(pi/((6*(x^2 + 1))))/5 - (pi*x^2)*cos(pi/((6*(x^2 + 1))))/(15*(x^2 + 1)^2)
Denominador racional
[src]
2
/ 2\ / pi \ 2 / pi \
15*\1 + x / *sin|--------| - 5*pi*x *cos|--------|
| 2| | 2|
\6 + 6*x / \6 + 6*x /
--------------------------------------------------
2
/ 2\
75*\1 + x /
$$\frac{- 5 \pi x^{2} \cos{\left(\frac{\pi}{6 x^{2} + 6} \right)} + 15 \left(x^{2} + 1\right)^{2} \sin{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{75 \left(x^{2} + 1\right)^{2}}$$
(15*(1 + x^2)^2*sin(pi/(6 + 6*x^2)) - 5*pi*x^2*cos(pi/(6 + 6*x^2)))/(75*(1 + x^2)^2)
Combinatoria
[src]
/ pi \ 4 / pi \ 2 / pi \ 2 / pi \
3*sin|--------| + 3*x *sin|--------| + 6*x *sin|--------| - pi*x *cos|--------|
| 2| | 2| | 2| | 2|
\6 + 6*x / \6 + 6*x / \6 + 6*x / \6 + 6*x /
-------------------------------------------------------------------------------
2
/ 2\
15*\1 + x /
$$\frac{3 x^{4} \sin{\left(\frac{\pi}{6 x^{2} + 6} \right)} + 6 x^{2} \sin{\left(\frac{\pi}{6 x^{2} + 6} \right)} - \pi x^{2} \cos{\left(\frac{\pi}{6 x^{2} + 6} \right)} + 3 \sin{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{15 \left(x^{2} + 1\right)^{2}}$$
(3*sin(pi/(6 + 6*x^2)) + 3*x^4*sin(pi/(6 + 6*x^2)) + 6*x^2*sin(pi/(6 + 6*x^2)) - pi*x^2*cos(pi/(6 + 6*x^2)))/(15*(1 + x^2)^2)
Unión de expresiones racionales
[src]
2
/ 2\ / pi \ 2 / pi \
3*\1 + x / *sin|--------| - pi*x *cos|--------|
| 2| | 2|
\6 + 6*x / \6 + 6*x /
-----------------------------------------------
2
/ 2\
15*\1 + x /
$$\frac{- \pi x^{2} \cos{\left(\frac{\pi}{6 x^{2} + 6} \right)} + 3 \left(x^{2} + 1\right)^{2} \sin{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{15 \left(x^{2} + 1\right)^{2}}$$
(3*(1 + x^2)^2*sin(pi/(6 + 6*x^2)) - pi*x^2*cos(pi/(6 + 6*x^2)))/(15*(1 + x^2)^2)
Potencias
[src]
/ pi*I -pi*I \
| -------- --------|
/ -pi*I pi*I \ | 2 2|
| -------- --------| | 6 + 6*x 6 + 6*x |
| 2 2| 2 |e e |
| 6 + 6*x 6 + 6*x | pi*x *|--------- + ---------|
I*\- e + e / \ 2 2 /
- --------------------------- - -----------------------------
10 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \left(\frac{e^{\frac{i \pi}{6 x^{2} + 6}}}{2} + \frac{e^{- \frac{i \pi}{6 x^{2} + 6}}}{2}\right)}{15 \left(x^{2} + 1\right)^{2}} - \frac{i \left(e^{\frac{i \pi}{6 x^{2} + 6}} - e^{- \frac{i \pi}{6 x^{2} + 6}}\right)}{10}$$
/ pi \ 2 / pi \
sin|--------| pi*x *cos|--------|
| 2| | 2|
\6 + 6*x / \6 + 6*x /
------------- - -------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \cos{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{5}$$
sin(pi/(6 + 6*x^2))/5 - pi*x^2*cos(pi/(6 + 6*x^2))/(15*(1 + x^2)^2)
Respuesta numérica
[src]
0.2*sin(pi/((6*(x^2 + 1)))) - 0.20943951023932*x^2*cos(pi/((6*(x^2 + 1))))/(1.0 + x^2)^2
0.2*sin(pi/((6*(x^2 + 1)))) - 0.20943951023932*x^2*cos(pi/((6*(x^2 + 1))))/(1.0 + x^2)^2
Denominador común
[src]
/ pi \ 2 / pi \
sin|----------| pi*x *cos|----------|
| / 2\| | / 2\|
\6*\1 + x // \6*\1 + x //
--------------- - ---------------------
5 4 2
15 + 15*x + 30*x
$$- \frac{\pi x^{2} \cos{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{15 x^{4} + 30 x^{2} + 15} + \frac{\sin{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{5}$$
sin(pi/(6*(1 + x^2)))/5 - pi*x^2*cos(pi/(6*(1 + x^2)))/(15 + 15*x^4 + 30*x^2)
Parte trigonométrica
[src]
/ pi \ 2 / 2/ pi \\
2*tan|------------| pi*x *|1 - tan |------------||
| / 2\| | | / 2\||
\2*\6 + 6*x // \ \2*\6 + 6*x ///
-------------------------- - -------------------------------------
/ 2/ pi \\ 2
5*|1 + tan |------------|| / 2\ / 2/ pi \\
| | / 2\|| 15*\1 + x / *|1 + tan |------------||
\ \2*\6 + 6*x /// | | / 2\||
\ \2*\6 + 6*x ///
$$- \frac{\pi x^{2} \left(1 - \tan^{2}{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)}\right)}{15 \left(x^{2} + 1\right)^{2} \left(\tan^{2}{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)}}{5 \left(\tan^{2}{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)} + 1\right)}$$
/ pi \ 2 / 2/ pi \\
2*cot|------------| pi*x *|-1 + cot |------------||
| / 2\| | | / 2\||
\2*\6 + 6*x // \ \2*\6 + 6*x ///
-------------------------- - -------------------------------------
/ 2/ pi \\ 2
5*|1 + cot |------------|| / 2\ / 2/ pi \\
| | / 2\|| 15*\1 + x / *|1 + cot |------------||
\ \2*\6 + 6*x /// | | / 2\||
\ \2*\6 + 6*x ///
$$- \frac{\pi x^{2} \left(\cot^{2}{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)} - 1\right)}{15 \left(x^{2} + 1\right)^{2} \left(\cot^{2}{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)}}{5 \left(\cot^{2}{\left(\frac{\pi}{2 \left(6 x^{2} + 6\right)} \right)} + 1\right)}$$
2 / 2 \
pi*x *|1 - -----------------|
| 2/ pi \|
| csc |-----------||
| | / 2\||
2 \ \12*\1 + x ///
---------------------------------------- - -----------------------------
/ pi \ /pi pi \ 2
5*csc|-----------|*csc|-- - -----------| / 2\
| / 2\| |2 / 2\| 15*\1 + x /
\12*\1 + x // \ 12*\1 + x //
$$- \frac{\pi x^{2} \left(1 - \frac{2}{\csc^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{2}{5 \csc{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} \csc{\left(\frac{\pi}{2} - \frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}$$
2
1 pi*x
---------------------- - --------------------------
/ pi pi \ 2
5*sec|- -- + --------| / 2\ / pi \
| 2 2| 15*\1 + x / *sec|--------|
\ 6 + 6*x / | 2|
\6 + 6*x /
$$- \frac{\pi x^{2}}{15 \left(x^{2} + 1\right)^{2} \sec{\left(\frac{\pi}{6 x^{2} + 6} \right)}} + \frac{1}{5 \sec{\left(- \frac{\pi}{2} + \frac{\pi}{6 x^{2} + 6} \right)}}$$
/ pi \ /pi pi \ 2 / 2/ pi \\
2*sin|-----------|*sin|-- + -----------| pi*x *|1 - 2*sin |-----------||
| / 2\| |2 / 2\| | | / 2\||
\12*\1 + x // \ 12*\1 + x // \ \12*\1 + x ///
---------------------------------------- - -------------------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \left(1 - 2 \sin^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{2 \sin{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} \sin{\left(\frac{\pi}{2} + \frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}{5}$$
/ pi pi \ 2 / pi \
cos|- -- + --------| pi*x *cos|--------|
| 2 2| | 2|
\ 6 + 6*x / \6 + 6*x /
-------------------- - -------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \cos{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\pi}{6 x^{2} + 6} \right)}}{5}$$
/ pi \ / pi \ 2 / 2/ pi \\
2*cos|-----------|*sin|-----------| pi*x *|1 - 2*sin |-----------||
| / 2\| | / 2\| | | / 2\||
\12*\1 + x // \12*\1 + x // \ \12*\1 + x ///
----------------------------------- - -------------------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \left(1 - 2 \sin^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{2 \sin{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} \cos{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}{5}$$
2
1 pi*x
------------------------ - ----------------------------
/ pi pi \ 2
5*sec|- -- + ----------| / 2\ / pi \
| 2 / 2\| 15*\1 + x / *sec|----------|
\ 6*\1 + x // | / 2\|
\6*\1 + x //
$$- \frac{\pi x^{2}}{15 \left(x^{2} + 1\right)^{2} \sec{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}} + \frac{1}{5 \sec{\left(- \frac{\pi}{2} + \frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}$$
/ pi \ 2 /pi pi \
sin|----------| pi*x *sin|-- + ----------|
| / 2\| |2 / 2\|
\6*\1 + x // \ 6*\1 + x //
--------------- - --------------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \sin{\left(\frac{\pi}{2} + \frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{5}$$
/ 2/ pi \ \
| 8*cot |-----------| |
| | / 2\| |
2 | \24*\1 + x // |
pi*x *|1 - ------------------------|
| 2|
/ 2/ pi \\ / pi \ | / 2/ pi \\ |
4*|-1 + cot |-----------||*cot|-----------| | |1 + cot |-----------|| |
| | / 2\|| | / 2\| | | | / 2\|| |
\ \24*\1 + x /// \24*\1 + x // \ \ \24*\1 + x /// /
------------------------------------------- - ------------------------------------
2 2
/ 2/ pi \\ / 2\
5*|1 + cot |-----------|| 15*\1 + x /
| | / 2\||
\ \24*\1 + x ///
$$- \frac{\pi x^{2} \left(1 - \frac{8 \cot^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)}}{\left(\cot^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)} + 1\right)^{2}}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)} - 1\right) \cot{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)}}{5 \left(\cot^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)} + 1\right)^{2}}$$
/ pi \ 2 /pi pi \
sin|--------| pi*x *sin|-- + --------|
| 2| |2 2|
\6 + 6*x / \ 6 + 6*x /
------------- - ------------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \sin{\left(\frac{\pi}{2} + \frac{\pi}{6 x^{2} + 6} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{5}$$
2
1 pi*x
----------------- - ---------------------------------
/ pi \ 2
5*csc|----------| / 2\ /pi pi \
| / 2\| 15*\1 + x / *csc|-- - ----------|
\6*\1 + x // |2 / 2\|
\ 6*\1 + x //
$$- \frac{\pi x^{2}}{15 \left(x^{2} + 1\right)^{2} \csc{\left(\frac{\pi}{2} - \frac{\pi}{6 \left(x^{2} + 1\right)} \right)}} + \frac{1}{5 \csc{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}$$
2
1 pi*x
--------------- - -------------------------------
/ pi \ 2
5*csc|--------| / 2\ /pi pi \
| 2| 15*\1 + x / *csc|-- - --------|
\6 + 6*x / |2 2|
\ 6 + 6*x /
$$- \frac{\pi x^{2}}{15 \left(x^{2} + 1\right)^{2} \csc{\left(\frac{\pi}{2} - \frac{\pi}{6 x^{2} + 6} \right)}} + \frac{1}{5 \csc{\left(\frac{\pi}{6 x^{2} + 6} \right)}}$$
/ pi \ 2 / 2/ pi \\
2*tan|-----------| pi*x *|1 - tan |-----------||
| / 2\| | | / 2\||
\12*\1 + x // \ \12*\1 + x ///
------------------------- - ------------------------------------
/ 2/ pi \\ 2
5*|1 + tan |-----------|| / 2\ / 2/ pi \\
| | / 2\|| 15*\1 + x / *|1 + tan |-----------||
\ \12*\1 + x /// | | / 2\||
\ \12*\1 + x ///
$$- \frac{\pi x^{2} \left(1 - \tan^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}\right)}{15 \left(x^{2} + 1\right)^{2} \left(\tan^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}{5 \left(\tan^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} + 1\right)}$$
/ pi \ 2 / pi \
sin|--------| pi*x *cos|--------|
| 2| | 2|
\6 + 6*x / \6 + 6*x /
------------- - -------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \cos{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\sin{\left(\frac{\pi}{6 x^{2} + 6} \right)}}{5}$$
/ 2/ pi \ \
| 8*tan |-----------| |
| | / 2\| |
2 | \24*\1 + x // |
pi*x *|1 - ------------------------|
| 2|
/ 2/ pi \\ / pi \ | / 2/ pi \\ |
4*|1 - tan |-----------||*tan|-----------| | |1 + tan |-----------|| |
| | / 2\|| | / 2\| | | | / 2\|| |
\ \24*\1 + x /// \24*\1 + x // \ \ \24*\1 + x /// /
------------------------------------------ - ------------------------------------
2 2
/ 2/ pi \\ / 2\
5*|1 + tan |-----------|| 15*\1 + x /
| | / 2\||
\ \24*\1 + x ///
$$- \frac{\pi x^{2} \left(1 - \frac{8 \tan^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)}}{\left(\tan^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)} + 1\right)^{2}}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)}\right) \tan{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)}}{5 \left(\tan^{2}{\left(\frac{\pi}{24 \left(x^{2} + 1\right)} \right)} + 1\right)^{2}}$$
/ pi pi \ 2 / pi \
cos|- -- + ----------| pi*x *cos|----------|
| 2 / 2\| | / 2\|
\ 6*\1 + x // \6*\1 + x //
---------------------- - ---------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \cos{\left(\frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{15 \left(x^{2} + 1\right)^{2}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\pi}{6 \left(x^{2} + 1\right)} \right)}}{5}$$
/ pi \ 2 / 2/ pi \\
2*cot|-----------| pi*x *|-1 + cot |-----------||
| / 2\| | | / 2\||
\12*\1 + x // \ \12*\1 + x ///
------------------------- - ------------------------------------
/ 2/ pi \\ 2
5*|1 + cot |-----------|| / 2\ / 2/ pi \\
| | / 2\|| 15*\1 + x / *|1 + cot |-----------||
\ \12*\1 + x /// | | / 2\||
\ \12*\1 + x ///
$$- \frac{\pi x^{2} \left(\cot^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} - 1\right)}{15 \left(x^{2} + 1\right)^{2} \left(\cot^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}{5 \left(\cot^{2}{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} + 1\right)}$$
2
1 pi*x
--------------- - --------------------------
/ pi \ 2
5*csc|--------| / 2\ / pi \
| 2| 15*\1 + x / *sec|--------|
\6 + 6*x / | 2|
\6 + 6*x /
$$- \frac{\pi x^{2}}{15 \left(x^{2} + 1\right)^{2} \sec{\left(\frac{\pi}{6 x^{2} + 6} \right)}} + \frac{1}{5 \csc{\left(\frac{\pi}{6 x^{2} + 6} \right)}}$$
/ pi \ / pi pi \ 2 / 2/ pi pi \\
2*cos|-----------|*cos|- -- + -----------| pi*x *|1 - 2*cos |- -- + -----------||
| / 2\| | 2 / 2\| | | 2 / 2\||
\12*\1 + x // \ 12*\1 + x // \ \ 12*\1 + x ///
------------------------------------------ - --------------------------------------
5 2
/ 2\
15*\1 + x /
$$- \frac{\pi x^{2} \left(1 - 2 \cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{12 \left(x^{2} + 1\right)} \right)}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{2 \cos{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} \cos{\left(- \frac{\pi}{2} + \frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}{5}$$
2 / 2 \
pi*x *|1 - ------------------------|
| 2/ pi pi \|
| sec |- -- + -----------||
| | 2 / 2\||
2 \ \ 12*\1 + x ///
------------------------------------------ - ------------------------------------
/ pi \ / pi pi \ 2
5*sec|-----------|*sec|- -- + -----------| / 2\
| / 2\| | 2 / 2\| 15*\1 + x /
\12*\1 + x // \ 12*\1 + x //
$$- \frac{\pi x^{2} \left(1 - \frac{2}{\sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}\right)}{15 \left(x^{2} + 1\right)^{2}} + \frac{2}{5 \sec{\left(\frac{\pi}{12 \left(x^{2} + 1\right)} \right)} \sec{\left(- \frac{\pi}{2} + \frac{\pi}{12 \left(x^{2} + 1\right)} \right)}}$$
2/(5*sec(pi/(12*(1 + x^2)))*sec(-pi/2 + pi/(12*(1 + x^2)))) - pi*x^2*(1 - 2/sec(-pi/2 + pi/(12*(1 + x^2)))^2)/(15*(1 + x^2)^2)