Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(6*x)/((2*sin(3*x)))+2*sin(33/2)^2*x-sin(6)^2*x-cos(6)^2*x
¿Cómo vas a descomponer esta sin(6*x)/((2*sin(3*x)))+2*sin(33/2)^2*x-sin(6)^2*x-cos(6)^2*x expresión en fracciones?
Solución
Ha introducido
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sin(6*x) 2 2 2 ---------- + 2*sin (33/2)*x - sin (6)*x - cos (6)*x 2*sin(3*x)
$$- x \cos^{2}{\left(6 \right)} + \left(- x \sin^{2}{\left(6 \right)} + \left(x 2 \sin^{2}{\left(\frac{33}{2} \right)} + \frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}\right)\right)$$
sin(6*x)/((2*sin(3*x))) + (2*sin(33/2)^2)*x - sin(6)^2*x - cos(6)^2*x
Simplificación general
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-x*cos(33) + cos(3*x)
$$- x \cos{\left(33 \right)} + \cos{\left(3 x \right)}$$
-x*cos(33) + cos(3*x)
Denominador común
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sin(6*x) 2 2 2 ---------- - x*cos (6) - x*sin (6) + 2*x*sin (33/2) 2*sin(3*x)
$$- x \cos^{2}{\left(6 \right)} - x \sin^{2}{\left(6 \right)} + 2 x \sin^{2}{\left(\frac{33}{2} \right)} + \frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
sin(6*x)/(2*sin(3*x)) - x*cos(6)^2 - x*sin(6)^2 + 2*x*sin(33/2)^2
Potencias
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sin(6*x) 2 2 2 ---------- - x*cos (6) - x*sin (6) + 2*x*sin (33/2) 2*sin(3*x)
$$- x \cos^{2}{\left(6 \right)} - x \sin^{2}{\left(6 \right)} + 2 x \sin^{2}{\left(\frac{33}{2} \right)} + \frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
2
/ -33*I 33*I\
2 | ----- ----| 2
-6*I*x 6*I*x / -6*I 6*I\ | 2 2 | / -6*I 6*I\
- e + e |e e | x*\- e + e / x*\- e + e /
---------------------- - x*|----- + ----| - --------------------- + -------------------
/ -3*I*x 3*I*x\ \ 2 2 / 2 4
2*\- e + e /
$$- x \left(\frac{e^{6 i}}{2} + \frac{e^{- 6 i}}{2}\right)^{2} + \frac{x \left(- e^{- 6 i} + e^{6 i}\right)^{2}}{4} - \frac{x \left(e^{\frac{33 i}{2}} - e^{- \frac{33 i}{2}}\right)^{2}}{2} + \frac{e^{6 i x} - e^{- 6 i x}}{2 \left(e^{3 i x} - e^{- 3 i x}\right)}$$
(-exp(-6*i*x) + exp(6*i*x))/(2*(-exp(-3*i*x) + exp(3*i*x))) - x*(exp(-6*i)/2 + exp(6*i)/2)^2 - x*(-exp(-33*i/2) + exp(33*i/2))^2/2 + x*(-exp(-6*i) + exp(6*i))^2/4
Unión de expresiones racionales
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2 2 2
- 2*x*cos (6)*sin(3*x) - 2*x*sin (6)*sin(3*x) + 4*x*sin (33/2)*sin(3*x) + sin(6*x)
----------------------------------------------------------------------------------
2*sin(3*x)
$$\frac{- 2 x \sin{\left(3 x \right)} \cos^{2}{\left(6 \right)} - 2 x \sin^{2}{\left(6 \right)} \sin{\left(3 x \right)} + 4 x \sin^{2}{\left(\frac{33}{2} \right)} \sin{\left(3 x \right)} + \sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
(-2*x*cos(6)^2*sin(3*x) - 2*x*sin(6)^2*sin(3*x) + 4*x*sin(33/2)^2*sin(3*x) + sin(6*x))/(2*sin(3*x))
Compilar la expresión
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sin(6*x) 2 2 2 ---------- - x*cos (6) - x*sin (6) + 2*x*sin (33/2) 2*sin(3*x)
$$- x \cos^{2}{\left(6 \right)} - x \sin^{2}{\left(6 \right)} + 2 x \sin^{2}{\left(\frac{33}{2} \right)} + \frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
sin(6*x)/(2*sin(3*x)) - x*cos(6)^2 - x*sin(6)^2 + 2*x*sin(33/2)^2
Denominador racional
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2 2 2
- 2*x*cos (6)*sin(3*x) - 2*x*sin (6)*sin(3*x) + 4*x*sin (33/2)*sin(3*x) + sin(6*x)
----------------------------------------------------------------------------------
2*sin(3*x)
$$\frac{- 2 x \sin{\left(3 x \right)} \cos^{2}{\left(6 \right)} - 2 x \sin^{2}{\left(6 \right)} \sin{\left(3 x \right)} + 4 x \sin^{2}{\left(\frac{33}{2} \right)} \sin{\left(3 x \right)} + \sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
(-2*x*cos(6)^2*sin(3*x) - 2*x*sin(6)^2*sin(3*x) + 4*x*sin(33/2)^2*sin(3*x) + sin(6*x))/(2*sin(3*x))
Combinatoria
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/ 2 2 2 \
-\-sin(6*x) - 4*x*sin (33/2)*sin(3*x) + 2*x*cos (6)*sin(3*x) + 2*x*sin (6)*sin(3*x)/
-------------------------------------------------------------------------------------
2*sin(3*x)
$$- \frac{- 4 x \sin^{2}{\left(\frac{33}{2} \right)} \sin{\left(3 x \right)} + 2 x \sin^{2}{\left(6 \right)} \sin{\left(3 x \right)} + 2 x \sin{\left(3 x \right)} \cos^{2}{\left(6 \right)} - \sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
-(-sin(6*x) - 4*x*sin(33/2)^2*sin(3*x) + 2*x*cos(6)^2*sin(3*x) + 2*x*sin(6)^2*sin(3*x))/(2*sin(3*x))
Parte trigonométrica
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/ 2 \
x*\1 - tan (33/2)/
- ------------------ + cos(3*x)
2
1 + tan (33/2)
$$- \frac{x \left(1 - \tan^{2}{\left(\frac{33}{2} \right)}\right)}{1 + \tan^{2}{\left(\frac{33}{2} \right)}} + \cos{\left(3 x \right)}$$
csc(3*x) x x 2*x
---------- - ------- - ------- + ----------
2*csc(6*x) 2 2 2
csc (6) sec (6) csc (33/2)
$$- \frac{x}{\sec^{2}{\left(6 \right)}} - \frac{x}{\csc^{2}{\left(6 \right)}} + \frac{2 x}{\csc^{2}{\left(\frac{33}{2} \right)}} + \frac{\csc{\left(3 x \right)}}{2 \csc{\left(6 x \right)}}$$
/ pi\
sec|3*x - --|
\ 2 / x x 2*x
--------------- - ------- - ------------ + -------------
/ pi\ 2 2/ pi\ 2/33 pi\
2*sec|6*x - --| sec (6) sec |6 - --| sec |-- - --|
\ 2 / \ 2 / \2 2 /
$$- \frac{x}{\sec^{2}{\left(6 \right)}} - \frac{x}{\sec^{2}{\left(6 - \frac{\pi}{2} \right)}} + \frac{2 x}{\sec^{2}{\left(\frac{33}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(3 x - \frac{\pi}{2} \right)}}{2 \sec{\left(6 x - \frac{\pi}{2} \right)}}$$
2 / 2/3*x\\
/ 2 \ 2 2 |1 + cot |---||*cot(3*x)
x*\-1 + cot (3)/ 4*x*cot (3) 8*x*cot (33/4) \ \ 2 //
- ----------------- - -------------- + ----------------- + --------------------------
2 2 2 / 2 \ /3*x\
/ 2 \ / 2 \ / 2 \ 2*\1 + cot (3*x)/*cot|---|
\1 + cot (3)/ \1 + cot (3)/ \1 + cot (33/4)/ \ 2 /
$$- \frac{x \left(-1 + \cot^{2}{\left(3 \right)}\right)^{2}}{\left(1 + \cot^{2}{\left(3 \right)}\right)^{2}} - \frac{4 x \cot^{2}{\left(3 \right)}}{\left(1 + \cot^{2}{\left(3 \right)}\right)^{2}} + \frac{8 x \cot^{2}{\left(\frac{33}{4} \right)}}{\left(\cot^{2}{\left(\frac{33}{4} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \cot{\left(3 x \right)}}{2 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(\frac{3 x}{2} \right)}}$$
1 x -------- - ------- sec(3*x) sec(33)
$$- \frac{x}{\sec{\left(33 \right)}} + \frac{1}{\sec{\left(3 x \right)}}$$
/ pi\
cos|6*x - --|
\ 2 / 2 2/ pi\ 2/33 pi\
--------------- - x*cos (6) - x*cos |6 - --| + 2*x*cos |-- - --|
/ pi\ \ 2 / \2 2 /
2*cos|3*x - --|
\ 2 /
$$- x \cos^{2}{\left(6 \right)} - x \cos^{2}{\left(6 - \frac{\pi}{2} \right)} + 2 x \cos^{2}{\left(\frac{33}{2} - \frac{\pi}{2} \right)} + \frac{\cos{\left(6 x - \frac{\pi}{2} \right)}}{2 \cos{\left(3 x - \frac{\pi}{2} \right)}}$$
1 2*x
x + -------- - ----------
sec(3*x) 2
sec (33/2)
$$- \frac{2 x}{\sec^{2}{\left(\frac{33}{2} \right)}} + x + \frac{1}{\sec{\left(3 x \right)}}$$
2/33 pi\ /pi \
x - 2*x*sin |-- + --| + sin|-- + 3*x|
\2 2 / \2 /
$$- 2 x \sin^{2}{\left(\frac{\pi}{2} + \frac{33}{2} \right)} + x + \sin{\left(3 x + \frac{\pi}{2} \right)}$$
2/3*x\
-1 + cot |---| / 2 \
\ 2 / x*\-1 + cot (33/2)/
-------------- - -------------------
2/3*x\ 2
1 + cot |---| 1 + cot (33/2)
\ 2 /
$$- \frac{x \left(-1 + \cot^{2}{\left(\frac{33}{2} \right)}\right)}{\cot^{2}{\left(\frac{33}{2} \right)} + 1} + \frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1}$$
2/3*x\ 2
1 - tan |---| / 2 \
\ 2 / 2*x*\1 - tan (33/4)/
x + ------------- - ---------------------
2/3*x\ 2
1 + tan |---| / 2 \
\ 2 / \1 + tan (33/4)/
$$- \frac{2 x \left(1 - \tan^{2}{\left(\frac{33}{4} \right)}\right)^{2}}{\left(1 + \tan^{2}{\left(\frac{33}{4} \right)}\right)^{2}} + x + \frac{1 - \tan^{2}{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1}$$
-x*cos(33) + cos(3*x)
$$- x \cos{\left(33 \right)} + \cos{\left(3 x \right)}$$
1 x ------------- - ------------- /pi \ / pi\ csc|-- - 3*x| csc|-33 + --| \2 / \ 2 /
$$- \frac{x}{\csc{\left(-33 + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 3 x + \frac{\pi}{2} \right)}}$$
2 / 2/3*x\\
/ 2 \ 2 2 |1 + tan |---||*tan(3*x)
x*\1 - tan (3)/ 4*x*tan (3) 8*x*tan (33/4) \ \ 2 //
- ---------------- - -------------- + ----------------- + --------------------------
2 2 2 / 2 \ /3*x\
/ 2 \ / 2 \ / 2 \ 2*\1 + tan (3*x)/*tan|---|
\1 + tan (3)/ \1 + tan (3)/ \1 + tan (33/4)/ \ 2 /
$$- \frac{x \left(1 - \tan^{2}{\left(3 \right)}\right)^{2}}{\left(\tan^{2}{\left(3 \right)} + 1\right)^{2}} - \frac{4 x \tan^{2}{\left(3 \right)}}{\left(\tan^{2}{\left(3 \right)} + 1\right)^{2}} + \frac{8 x \tan^{2}{\left(\frac{33}{4} \right)}}{\left(1 + \tan^{2}{\left(\frac{33}{4} \right)}\right)^{2}} + \frac{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \tan{\left(3 x \right)}}{2 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(\frac{3 x}{2} \right)}}$$
sin(6*x) 2 2/ pi\ 2 ---------- - x*sin (6) - x*sin |6 + --| + 2*x*sin (33/2) 2*sin(3*x) \ 2 /
$$- x \sin^{2}{\left(\frac{\pi}{2} + 6 \right)} - x \sin^{2}{\left(6 \right)} + 2 x \sin^{2}{\left(\frac{33}{2} \right)} + \frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
2/3*x\ 2
-1 + cot |---| / 2 \
\ 2 / 2*x*\-1 + cot (33/4)/
x + -------------- - ----------------------
2/3*x\ 2
1 + cot |---| / 2 \
\ 2 / \1 + cot (33/4)/
$$- \frac{2 x \left(-1 + \cot^{2}{\left(\frac{33}{4} \right)}\right)^{2}}{\left(\cot^{2}{\left(\frac{33}{4} \right)} + 1\right)^{2}} + x + \frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1}$$
/ pi\ /pi \
- x*sin|33 + --| + sin|-- + 3*x|
\ 2 / \2 /
$$- x \sin{\left(\frac{\pi}{2} + 33 \right)} + \sin{\left(3 x + \frac{\pi}{2} \right)}$$
2 x - 2*x*cos (33/2) + cos(3*x)
$$- 2 x \cos^{2}{\left(\frac{33}{2} \right)} + x + \cos{\left(3 x \right)}$$
csc(3*x) x x 2*x
---------- - ------- - ------------- + ----------
2*csc(6*x) 2 2/ pi\ 2
csc (6) csc |-6 + --| csc (33/2)
\ 2 /
$$- \frac{x}{\csc^{2}{\left(-6 + \frac{\pi}{2} \right)}} - \frac{x}{\csc^{2}{\left(6 \right)}} + \frac{2 x}{\csc^{2}{\left(\frac{33}{2} \right)}} + \frac{\csc{\left(3 x \right)}}{2 \csc{\left(6 x \right)}}$$
2/3*x\
1 - tan |---| / 2 \
\ 2 / x*\1 - tan (33/2)/
------------- - ------------------
2/3*x\ 2
1 + tan |---| 1 + tan (33/2)
\ 2 /
$$- \frac{x \left(1 - \tan^{2}{\left(\frac{33}{2} \right)}\right)}{1 + \tan^{2}{\left(\frac{33}{2} \right)}} + \frac{1 - \tan^{2}{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1}$$
sin(6*x) 2 2 2 ---------- - x*cos (6) - x*sin (6) + 2*x*sin (33/2) 2*sin(3*x)
$$- x \cos^{2}{\left(6 \right)} - x \sin^{2}{\left(6 \right)} + 2 x \sin^{2}{\left(\frac{33}{2} \right)} + \frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$
1 2*x
x + ------------- - ---------------
/pi \ 2/ 33 pi\
csc|-- - 3*x| csc |- -- + --|
\2 / \ 2 2 /
$$- \frac{2 x}{\csc^{2}{\left(- \frac{33}{2} + \frac{\pi}{2} \right)}} + x + \frac{1}{\csc{\left(- 3 x + \frac{\pi}{2} \right)}}$$
x + 1/csc(pi/2 - 3*x) - 2*x/csc(-33/2 + pi/2)^2
Respuesta numérica
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0.0132767472230596*x + 0.5*sin(6*x)/sin(3*x)
0.0132767472230596*x + 0.5*sin(6*x)/sin(3*x)