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