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