Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
atan(x)^(1/3)*cos(x)+sin(x)/(3*(1+x^2)*atan(x)^(2/3))
¿Cómo vas a descomponer esta atan(x)^(1/3)*cos(x)+sin(x)/(3*(1+x^2)*atan(x)^(2/3)) expresión en fracciones?
Solución
Ha introducido
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3 _________ sin(x)
\/ atan(x) *cos(x) + ---------------------
/ 2\ 2/3
3*\1 + x /*atan (x)
$$\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
atan(x)^(1/3)*cos(x) + sin(x)/(((3*(1 + x^2))*atan(x)^(2/3)))
Simplificación general
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sin(x) / 2\
------ + \1 + x /*atan(x)*cos(x)
3
--------------------------------
/ 2\ 2/3
\1 + x /*atan (x)
$$\frac{\left(x^{2} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(x \right)}}{3}}{\left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
(sin(x)/3 + (1 + x^2)*atan(x)*cos(x))/((1 + x^2)*atan(x)^(2/3))
Respuesta numérica
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atan(x)^0.333333333333333*cos(x) + atan(x)^(-0.666666666666667)*sin(x)/(3.0 + 3.0*x^2)
atan(x)^0.333333333333333*cos(x) + atan(x)^(-0.666666666666667)*sin(x)/(3.0 + 3.0*x^2)
Denominador común
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2
3*atan(x)*cos(x) + 3*x *atan(x)*cos(x) + sin(x)
-----------------------------------------------
2/3 2 2/3
3*atan (x) + 3*x *atan (x)
$$\frac{3 x^{2} \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)} + 3 \cos{\left(x \right)} \operatorname{atan}{\left(x \right)}}{3 x^{2} \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} + 3 \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
(3*atan(x)*cos(x) + 3*x^2*atan(x)*cos(x) + sin(x))/(3*atan(x)^(2/3) + 3*x^2*atan(x)^(2/3))
Unión de expresiones racionales
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/ 2\
3*\1 + x /*atan(x)*cos(x) + sin(x)
----------------------------------
/ 2\ 2/3
3*\1 + x /*atan (x)
$$\frac{3 \left(x^{2} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)}}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
(3*(1 + x^2)*atan(x)*cos(x) + sin(x))/(3*(1 + x^2)*atan(x)^(2/3))
Potencias
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3 _________ sin(x)
\/ atan(x) *cos(x) + ---------------------
/ 2\ 2/3
\3 + 3*x /*atan (x)
$$\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
/ I*x -I*x\ / -I*x I*x\
3 _________ |e e | I*\- e + e /
\/ atan(x) *|---- + -----| - -----------------------
\ 2 2 / / 2\ 2/3
2*\3 + 3*x /*atan (x)
$$\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \sqrt[3]{\operatorname{atan}{\left(x \right)}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
atan(x)^(1/3)*(exp(i*x)/2 + exp(-i*x)/2) - i*(-exp(-i*x) + exp(i*x))/(2*(3 + 3*x^2)*atan(x)^(2/3))
Combinatoria
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2
3*atan(x)*cos(x) + 3*x *atan(x)*cos(x) + sin(x)
-----------------------------------------------
/ 2\ 2/3
3*\1 + x /*atan (x)
$$\frac{3 x^{2} \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)} + 3 \cos{\left(x \right)} \operatorname{atan}{\left(x \right)}}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
(3*atan(x)*cos(x) + 3*x^2*atan(x)*cos(x) + sin(x))/(3*(1 + x^2)*atan(x)^(2/3))
Denominador racional
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/ 2\
\3 + 3*x /*atan(x)*cos(x) + sin(x)
----------------------------------
/ 2\ 2/3
\3 + 3*x /*atan (x)
$$\frac{\left(3 x^{2} + 3\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
((3 + 3*x^2)*atan(x)*cos(x) + sin(x))/((3 + 3*x^2)*atan(x)^(2/3))
Compilar la expresión
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3 _________ sin(x)
\/ atan(x) *cos(x) + ---------------------
/ 2\ 2/3
\3 + 3*x /*atan (x)
$$\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
atan(x)^(1/3)*cos(x) + sin(x)/((3 + 3*x^2)*atan(x)^(2/3))
Parte trigonométrica
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3 _________ 1
\/ atan(x) *cos(x) + ---------------------*sin(x)
/ 2\ 2/3
3*\1 + x /*atan (x)
$$\frac{1}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} \sin{\left(x \right)} + \cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}}$$
3 _________ sin(x)
\/ atan(x) *cos(x) + ---------------------
/ 2\ 2/3
\3 + 3*x /*atan (x)
$$\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
3 _________
\/ atan(x) 1
----------- + ----------------------------
sec(x) / 2\ 2/3
\3 + 3*x /*atan (x)*csc(x)
$$\frac{\sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\sec{\left(x \right)}} + \frac{1}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} \csc{\left(x \right)}}$$
3 _________ / pi\ sin(x)
\/ atan(x) *sin|x + --| + ---------------------
\ 2 / / 2\ 2/3
\3 + 3*x /*atan (x)
$$\sin{\left(x + \frac{\pi}{2} \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
/ pi\
cos|x - --|
3 _________ \ 2 /
\/ atan(x) *cos(x) + ---------------------
/ 2\ 2/3
\3 + 3*x /*atan (x)
$$\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
3 _________ / 2/x\\ /x\
\/ atan(x) *|-1 + cot |-|| 2*cot|-|
\ \2// \2/
-------------------------- + -----------------------------------
2/x\ / 2/x\\ / 2\ 2/3
1 + cot |-| |1 + cot |-||*\3 + 3*x /*atan (x)
\2/ \ \2//
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(3 x^{2} + 3\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
3 _________
\/ atan(x) 1
----------- + ---------------------------------
sec(x) / 2\ 2/3 / pi\
\3 + 3*x /*atan (x)*sec|x - --|
\ 2 /
$$\frac{\sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\sec{\left(x \right)}} + \frac{1}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
3 _________ \/ atan(x) 1 ----------- + ---------------------------- /pi \ / 2\ 2/3 csc|-- - x| \3 + 3*x /*atan (x)*csc(x) \2 /
$$\frac{\sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} \csc{\left(x \right)}}$$
3 _________ / 2/x\\ /x\
\/ atan(x) *|1 - tan |-|| 2*tan|-|
\ \2// \2/
------------------------- + -----------------------------------
2/x\ / 2/x\\ / 2\ 2/3
1 + tan |-| |1 + tan |-||*\3 + 3*x /*atan (x)
\2/ \ \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(3 x^{2} + 3\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}}$$
atan(x)^(1/3)*(1 - tan(x/2)^2)/(1 + tan(x/2)^2) + 2*tan(x/2)/((1 + tan(x/2)^2)*(3 + 3*x^2)*atan(x)^(2/3))