Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)/(a*sqrt(1-cos(x)^((2/a)^2)))
¿Cómo vas a descomponer esta sin(x)/(a*sqrt(1-cos(x)^((2/a)^2))) expresión en fracciones?
Solución
Ha introducido
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sin(x)
----------------------------
____________________
/ / 2\
/ |/2\ |
/ ||-| |
/ \\a/ /
a*\/ 1 - (cos(x))
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\left(\frac{2}{a}\right)^{2}}{\left(x \right)}}}$$
sin(x)/((a*sqrt(1 - cos(x)^((2/a)^2))))
Simplificación general
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sin(x)
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/ 4
/ --
/ 2
/ a
a*\/ 1 - (cos(x))
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}}$$
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
Potencias
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/ -I*x I*x\
-I*\- e + e /
-----------------------------------
______________________
/ 4
/ --
/ 2
/ a
/ / I*x -I*x\
/ |e e |
2*a* / 1 - |---- + -----|
\/ \ 2 2 /
$$- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 a \sqrt{1 - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\frac{4}{a^{2}}}}}$$
sin(x)
------------------------
________________
/ 4
/ --
/ 2
/ a
a*\/ 1 - (cos(x))
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}}$$
sin(x)
------------------------
_______________
/ 1
/ --
/ 2
/ a
/ 4
a*\/ 1 - cos (x)
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \left(\cos^{4}{\left(x \right)}\right)^{\frac{1}{a^{2}}}}}$$
sin(x)/(a*sqrt(1 - (cos(x)^4)^(a^(-2))))
Unión de expresiones racionales
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sin(x)
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________________
/ 4
/ --
/ 2
/ a
a*\/ 1 - (cos(x))
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}}$$
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
Combinatoria
[src]
sin(x)
--------------------------------------------------------------
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/ / 1 \ / 2 \ / 1 \
/ | --| | --| | --|
/ | 2| | 2| | 2|
/ | a | | a | | a |
a*\/ -\1 + (cos(x)) /*\1 + (cos(x)) /*\-1 + (cos(x)) /
$$\frac{\sin{\left(x \right)}}{a \sqrt{- \left(\cos^{\frac{1}{a^{2}}}{\left(x \right)} - 1\right) \left(\cos^{\frac{1}{a^{2}}}{\left(x \right)} + 1\right) \left(\cos^{\frac{2}{a^{2}}}{\left(x \right)} + 1\right)}}$$
sin(x)/(a*sqrt(-(1 + cos(x)^(a^(-2)))*(1 + cos(x)^(2/a^2))*(-1 + cos(x)^(a^(-2)))))
Denominador común
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sin(x)
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/ 4
/ --
/ 2
/ a
a*\/ 1 - (cos(x))
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}}$$
sin(x)/(a*sqrt(1 - cos(x)^(4/a^2)))
Denominador racional
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________________
/ 4
/ --
/ 2
/ a
-\/ 1 - (cos(x)) *sin(x)
-------------------------------
/ 4 \
| --|
| 2|
| a |
a*\-1 + (cos(x)) /
$$- \frac{\sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}} \sin{\left(x \right)}}{a \left(\cos^{\frac{4}{a^{2}}}{\left(x \right)} - 1\right)}$$
-sqrt(1 - cos(x)^(4/a^2))*sin(x)/(a*(-1 + cos(x)^(4/a^2)))
Parte trigonométrica
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/ pi\
cos|x - --|
\ 2 /
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________________
/ 4
/ --
/ 2
/ a
a*\/ 1 - (cos(x))
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}}$$
1
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/ 4
/ --
/ 2
/ a
/ / 1 \
a* / 1 - |-----------| *csc(x)
/ | /pi \|
/ |csc|-- - x||
\/ \ \2 //
$$\frac{1}{a \sqrt{1 - \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\frac{4}{a^{2}}}} \csc{\left(x \right)}}$$
1
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________________
/ 4
/ --
/ 2
/ a
/ / 1 \ / pi\
a* / 1 - |------| *sec|x - --|
\/ \sec(x)/ \ 2 /
$$\frac{1}{a \sqrt{1 - \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{4}{a^{2}}}} \sec{\left(x - \frac{\pi}{2} \right)}}$$
sin(x)
-------------------------------
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/ 4
/ --
/ 2
/ a
/ / / pi\\
a* / 1 - |sin|x + --||
\/ \ \ 2 //
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \sin^{\frac{4}{a^{2}}}{\left(x + \frac{\pi}{2} \right)}}}$$
/x\
2*cot|-|
\2/
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/ 4
/ --
/ 2
/ a
/ / 2/x\\
/ |-1 + cot |-||
/ 2/x\\ / | \2/|
a*|1 + cot |-||* / 1 - |------------|
\ \2// / | 2/x\ |
/ |1 + cot |-| |
\/ \ \2/ /
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{a \sqrt{1 - \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{4}{a^{2}}}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
1
---------------------------------
________________
/ 4
/ --
/ 2
/ a
/ / 1 \
a* / 1 - |------| *csc(x)
\/ \sec(x)/
$$\frac{1}{a \sqrt{1 - \left(\frac{1}{\sec{\left(x \right)}}\right)^{\frac{4}{a^{2}}}} \csc{\left(x \right)}}$$
sin(x)
------------------------
________________
/ 4
/ --
/ 2
/ a
a*\/ 1 - (cos(x))
$$\frac{\sin{\left(x \right)}}{a \sqrt{1 - \cos^{\frac{4}{a^{2}}}{\left(x \right)}}}$$
/x\
2*tan|-|
\2/
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/ 4
/ --
/ 2
/ a
/ / 2/x\\
/ |1 - tan |-||
/ 2/x\\ / | \2/|
a*|1 + tan |-||* / 1 - |-----------|
\ \2// / | 2/x\|
/ |1 + tan |-||
\/ \ \2//
$$\frac{2 \tan{\left(\frac{x}{2} \right)}}{a \sqrt{1 - \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{4}{a^{2}}}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2*tan(x/2)/(a*(1 + tan(x/2)^2)*sqrt(1 - ((1 - tan(x/2)^2)/(1 + tan(x/2)^2))^(4/a^2)))
Respuesta numérica
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(1.0 - cos(x)^(4.0/a^2))^(-0.5)*sin(x)/a
(1.0 - cos(x)^(4.0/a^2))^(-0.5)*sin(x)/a