Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin((a+1/2)*x)/(2*(a+1/2))+sin((1/2-a)*x)/(2*(1/2-a))
¿Cómo vas a descomponer esta sin((a+1/2)*x)/(2*(a+1/2))+sin((1/2-a)*x)/(2*(1/2-a)) expresión en fracciones?
Solución
Ha introducido
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sin((a + 1/2)*x) sin((1/2 - a)*x) ---------------- + ---------------- 2*(a + 1/2) 2*(1/2 - a)
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 \left(a + \frac{1}{2}\right)} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{2 \left(\frac{1}{2} - a\right)}$$
sin((a + 1/2)*x)/((2*(a + 1/2))) + sin((1/2 - a)*x)/((2*(1/2 - a)))
Simplificación general
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(1 + 2*a)*sin(x*(-1/2 + a)) + (-1 + 2*a)*sin(x*(1/2 + a))
---------------------------------------------------------
(1 + 2*a)*(-1 + 2*a)
$$\frac{\left(2 a - 1\right) \sin{\left(x \left(a + \frac{1}{2}\right) \right)} + \left(2 a + 1\right) \sin{\left(x \left(a - \frac{1}{2}\right) \right)}}{\left(2 a - 1\right) \left(2 a + 1\right)}$$
((1 + 2*a)*sin(x*(-1/2 + a)) + (-1 + 2*a)*sin(x*(1/2 + a)))/((1 + 2*a)*(-1 + 2*a))
Respuesta numérica
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sin((a + 1/2)*x)/(1.0 + 2.0*a) + sin((1/2 - a)*x)/(1.0 - 2.0*a)
sin((a + 1/2)*x)/(1.0 + 2.0*a) + sin((1/2 - a)*x)/(1.0 - 2.0*a)
Denominador racional
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/x \ / x \ /x \ / x \
- 4*sin|- + a*x| + 4*sin|- - + a*x| + 8*a*sin|- + a*x| + 8*a*sin|- - + a*x|
\2 / \ 2 / \2 / \ 2 /
---------------------------------------------------------------------------
(-2 + 4*a)*(2 + 4*a)
$$\frac{8 a \sin{\left(a x - \frac{x}{2} \right)} + 8 a \sin{\left(a x + \frac{x}{2} \right)} + 4 \sin{\left(a x - \frac{x}{2} \right)} - 4 \sin{\left(a x + \frac{x}{2} \right)}}{\left(4 a - 2\right) \left(4 a + 2\right)}$$
(-4*sin(x/2 + a*x) + 4*sin(-x/2 + a*x) + 8*a*sin(x/2 + a*x) + 8*a*sin(-x/2 + a*x))/((-2 + 4*a)*(2 + 4*a))
Denominador común
[src]
/x \ /x \ / x \ / x \
- sin|- + a*x| + 2*a*sin|- + a*x| + 2*a*sin|- - + a*x| + sin|- - + a*x|
\2 / \2 / \ 2 / \ 2 /
-----------------------------------------------------------------------
2
-1 + 4*a
$$\frac{2 a \sin{\left(a x - \frac{x}{2} \right)} + 2 a \sin{\left(a x + \frac{x}{2} \right)} + \sin{\left(a x - \frac{x}{2} \right)} - \sin{\left(a x + \frac{x}{2} \right)}}{4 a^{2} - 1}$$
(-sin(x/2 + a*x) + 2*a*sin(x/2 + a*x) + 2*a*sin(-x/2 + a*x) + sin(-x/2 + a*x))/(-1 + 4*a^2)
Potencias
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/ -I*x*(1/2 - a) I*x*(1/2 - a)\ / -I*x*(1/2 + a) I*x*(1/2 + a)\
I*\- e + e / I*\- e + e /
- -------------------------------------- - --------------------------------------
2*(1 - 2*a) 2*(1 + 2*a)
$$- \frac{i \left(e^{i x \left(a + \frac{1}{2}\right)} - e^{- i x \left(a + \frac{1}{2}\right)}\right)}{2 \left(2 a + 1\right)} - \frac{i \left(e^{i x \left(\frac{1}{2} - a\right)} - e^{- i x \left(\frac{1}{2} - a\right)}\right)}{2 \left(1 - 2 a\right)}$$
sin(x*(1/2 - a)) sin(x*(1/2 + a))
---------------- + ----------------
1 - 2*a 1 + 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{1 - 2 a}$$
sin(x*(1/2 - a))/(1 - 2*a) + sin(x*(1/2 + a))/(1 + 2*a)
Combinatoria
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/x \ /x \ / x \ / x \
- sin|- + a*x| + 2*a*sin|- + a*x| + 2*a*sin|- - + a*x| + sin|- - + a*x|
\2 / \2 / \ 2 / \ 2 /
-----------------------------------------------------------------------
(1 + 2*a)*(-1 + 2*a)
$$\frac{2 a \sin{\left(a x - \frac{x}{2} \right)} + 2 a \sin{\left(a x + \frac{x}{2} \right)} + \sin{\left(a x - \frac{x}{2} \right)} - \sin{\left(a x + \frac{x}{2} \right)}}{\left(2 a - 1\right) \left(2 a + 1\right)}$$
(-sin(x/2 + a*x) + 2*a*sin(x/2 + a*x) + 2*a*sin(-x/2 + a*x) + sin(-x/2 + a*x))/((1 + 2*a)*(-1 + 2*a))
Unión de expresiones racionales
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/x*(1 + 2*a)\ /x*(1 - 2*a)\
(1 - 2*a)*sin|-----------| + (1 + 2*a)*sin|-----------|
\ 2 / \ 2 /
-------------------------------------------------------
(1 - 2*a)*(1 + 2*a)
$$\frac{\left(1 - 2 a\right) \sin{\left(\frac{x \left(2 a + 1\right)}{2} \right)} + \left(2 a + 1\right) \sin{\left(\frac{x \left(1 - 2 a\right)}{2} \right)}}{\left(1 - 2 a\right) \left(2 a + 1\right)}$$
((1 - 2*a)*sin(x*(1 + 2*a)/2) + (1 + 2*a)*sin(x*(1 - 2*a)/2))/((1 - 2*a)*(1 + 2*a))
Compilar la expresión
[src]
sin((1/2 - a)*x) sin((a + 1/2)*x)
---------------- + ----------------
1 - 2*a 1 + 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{1 - 2 a}$$
sin((1/2 - a)*x)/(1 - 2*a) + sin((a + 1/2)*x)/(1 + 2*a)
Parte trigonométrica
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1 1 -------------------------- - --------------------------- (1 + 2*a)*csc(x*(1/2 + a)) (1 - 2*a)*csc(x*(-1/2 + a))
$$\frac{1}{\left(2 a + 1\right) \csc{\left(x \left(a + \frac{1}{2}\right) \right)}} - \frac{1}{\left(1 - 2 a\right) \csc{\left(x \left(a - \frac{1}{2}\right) \right)}}$$
/ pi \ / pi \
cos|- -- + x*(1/2 - a)| cos|- -- + x*(1/2 + a)|
\ 2 / \ 2 /
----------------------- + -----------------------
1 - 2*a 1 + 2*a
$$\frac{\cos{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}}{2 a + 1} + \frac{\cos{\left(x \left(\frac{1}{2} - a\right) - \frac{\pi}{2} \right)}}{1 - 2 a}$$
sin(x*(1/2 + a)) sin(x*(-1/2 + a))
---------------- - -----------------
1 + 2*a 1 - 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} - \frac{\sin{\left(x \left(a - \frac{1}{2}\right) \right)}}{1 - 2 a}$$
1 1 -------------------------- + -------------------------- (1 - 2*a)*csc(x*(1/2 - a)) (1 + 2*a)*csc(x*(1/2 + a))
$$\frac{1}{\left(2 a + 1\right) \csc{\left(x \left(a + \frac{1}{2}\right) \right)}} + \frac{1}{\left(1 - 2 a\right) \csc{\left(x \left(\frac{1}{2} - a\right) \right)}}$$
/ pi \ / pi \
cos|- -- + x*(1/2 + a)| cos|- -- + x*(-1/2 + a)|
\ 2 / \ 2 /
----------------------- - ------------------------
1 + 2*a 1 - 2*a
$$\frac{\cos{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}}{2 a + 1} - \frac{\cos{\left(x \left(a - \frac{1}{2}\right) - \frac{\pi}{2} \right)}}{1 - 2 a}$$
/x*(1/2 + a)\ /x*(1/2 - a)\
2*cot|-----------| 2*cot|-----------|
\ 2 / \ 2 /
--------------------------------- + ---------------------------------
/ 2/x*(1/2 + a)\\ / 2/x*(1/2 - a)\\
|1 + cot |-----------||*(1 + 2*a) |1 + cot |-----------||*(1 - 2*a)
\ \ 2 // \ \ 2 //
$$\frac{2 \cot{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\cot^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\cot^{2}{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)} + 1\right)}$$
/x*(-1/2 + a)\ /x*(1/2 + a)\
2*tan|------------| 2*tan|-----------|
\ 2 / \ 2 /
- ---------------------------------- + ---------------------------------
/ 2/x*(-1/2 + a)\\ / 2/x*(1/2 + a)\\
|1 + tan |------------||*(1 - 2*a) |1 + tan |-----------||*(1 + 2*a)
\ \ 2 // \ \ 2 //
$$\frac{2 \tan{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\tan^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\tan^{2}{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)} + 1\right)}$$
1 1
--------------------------------- + ---------------------------------
/ pi \ / pi \
(1 - 2*a)*sec|- -- + x*(1/2 - a)| (1 + 2*a)*sec|- -- + x*(1/2 + a)|
\ 2 / \ 2 /
$$\frac{1}{\left(2 a + 1\right) \sec{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - 2 a\right) \sec{\left(x \left(\frac{1}{2} - a\right) - \frac{\pi}{2} \right)}}$$
1 1 -----------*sin((1/2 - a)*x) + -----------*sin((a + 1/2)*x) 2*(1/2 - a) 2*(a + 1/2)
$$\frac{1}{2 \left(\frac{1}{2} - a\right)} \sin{\left(x \left(\frac{1}{2} - a\right) \right)} + \frac{1}{2 \left(a + \frac{1}{2}\right)} \sin{\left(x \left(a + \frac{1}{2}\right) \right)}$$
/x*(-1/2 + a)\ /x*(1/2 + a)\
2*cot|------------| 2*cot|-----------|
\ 2 / \ 2 /
- ---------------------------------- + ---------------------------------
/ 2/x*(-1/2 + a)\\ / 2/x*(1/2 + a)\\
|1 + cot |------------||*(1 - 2*a) |1 + cot |-----------||*(1 + 2*a)
\ \ 2 // \ \ 2 //
$$\frac{2 \cot{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\cot^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\cot^{2}{\left(\frac{x \left(a - \frac{1}{2}\right)}{2} \right)} + 1\right)}$$
sin(x*(1/2 - a)) sin(x*(1/2 + a))
---------------- + ----------------
1 - 2*a 1 + 2*a
$$\frac{\sin{\left(x \left(a + \frac{1}{2}\right) \right)}}{2 a + 1} + \frac{\sin{\left(x \left(\frac{1}{2} - a\right) \right)}}{1 - 2 a}$$
1 1
--------------------------------- - ----------------------------------
/ pi \ / pi \
(1 + 2*a)*sec|- -- + x*(1/2 + a)| (1 - 2*a)*sec|- -- + x*(-1/2 + a)|
\ 2 / \ 2 /
$$\frac{1}{\left(2 a + 1\right) \sec{\left(x \left(a + \frac{1}{2}\right) - \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - 2 a\right) \sec{\left(x \left(a - \frac{1}{2}\right) - \frac{\pi}{2} \right)}}$$
/x*(1/2 + a)\ /x*(1/2 - a)\
2*tan|-----------| 2*tan|-----------|
\ 2 / \ 2 /
--------------------------------- + ---------------------------------
/ 2/x*(1/2 + a)\\ / 2/x*(1/2 - a)\\
|1 + tan |-----------||*(1 + 2*a) |1 + tan |-----------||*(1 - 2*a)
\ \ 2 // \ \ 2 //
$$\frac{2 \tan{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)}}{\left(2 a + 1\right) \left(\tan^{2}{\left(\frac{x \left(a + \frac{1}{2}\right)}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)}}{\left(1 - 2 a\right) \left(\tan^{2}{\left(\frac{x \left(\frac{1}{2} - a\right)}{2} \right)} + 1\right)}$$
2*tan(x*(1/2 + a)/2)/((1 + tan(x*(1/2 + a)/2)^2)*(1 + 2*a)) + 2*tan(x*(1/2 - a)/2)/((1 + tan(x*(1/2 - a)/2)^2)*(1 - 2*a))