Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
atan(x^(1/3))*cos(x)+sin(x)/(3*x^(2/3)*(1+x^(2/3)))
¿Cómo vas a descomponer esta atan(x^(1/3))*cos(x)+sin(x)/(3*x^(2/3)*(1+x^(2/3))) expresión en fracciones?
Solución
Ha introducido
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/3 ___\ sin(x)
atan\\/ x /*cos(x) + -----------------
2/3 / 2/3\
3*x *\1 + x /
$$\cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \frac{\sin{\left(x \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
atan(x^(1/3))*cos(x) + sin(x)/(((3*x^(2/3))*(1 + x^(2/3))))
Simplificación general
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sin(x) 2/3 / 2/3\ /3 ___\
------ + x *\1 + x /*atan\\/ x /*cos(x)
3
-------------------------------------------
2/3 / 2/3\
x *\1 + x /
$$\frac{x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \frac{\sin{\left(x \right)}}{3}}{x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
(sin(x)/3 + x^(2/3)*(1 + x^(2/3))*atan(x^(1/3))*cos(x))/(x^(2/3)*(1 + x^(2/3)))
Respuesta numérica
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atan(x^(1/3))*cos(x) + 0.333333333333333*x^(-0.666666666666667)*sin(x)/(1.0 + x^0.666666666666667)
atan(x^(1/3))*cos(x) + 0.333333333333333*x^(-0.666666666666667)*sin(x)/(1.0 + x^0.666666666666667)
Denominador común
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sin(x) /3 ___\ --------------- + atan\\/ x /*cos(x) 2/3 4/3 3*x + 3*x
$$\cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \frac{\sin{\left(x \right)}}{3 x^{\frac{4}{3}} + 3 x^{\frac{2}{3}}}$$
sin(x)/(3*x^(2/3) + 3*x^(4/3)) + atan(x^(1/3))*cos(x)
Denominador racional
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2/3 / 2/3\ /3 ___\
3*x *\1 + x /*atan\\/ x /*cos(x) + sin(x)
---------------------------------------------
2/3 / 2/3\
3*x *\1 + x /
$$\frac{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \sin{\left(x \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
(3*x^(2/3)*(1 + x^(2/3))*atan(x^(1/3))*cos(x) + sin(x))/(3*x^(2/3)*(1 + x^(2/3)))
Unión de expresiones racionales
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2/3 / 2/3\ /3 ___\
3*x *\1 + x /*atan\\/ x /*cos(x) + sin(x)
---------------------------------------------
2/3 / 2/3\
3*x *\1 + x /
$$\frac{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \sin{\left(x \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
(3*x^(2/3)*(1 + x^(2/3))*atan(x^(1/3))*cos(x) + sin(x))/(3*x^(2/3)*(1 + x^(2/3)))
Combinatoria
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2/3 /3 ___\ 4/3 /3 ___\
3*x *atan\\/ x /*cos(x) + 3*x *atan\\/ x /*cos(x) + sin(x)
--------------------------------------------------------------
2/3 / 2/3\
3*x *\1 + x /
$$\frac{3 x^{\frac{4}{3}} \cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + 3 x^{\frac{2}{3}} \cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \sin{\left(x \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
(3*x^(2/3)*atan(x^(1/3))*cos(x) + 3*x^(4/3)*atan(x^(1/3))*cos(x) + sin(x))/(3*x^(2/3)*(1 + x^(2/3)))
Potencias
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/3 ___\ sin(x)
atan\\/ x /*cos(x) + -----------------
2/3 / 2/3\
x *\3 + 3*x /
$$\cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \frac{\sin{\left(x \right)}}{x^{\frac{2}{3}} \left(3 x^{\frac{2}{3}} + 3\right)}$$
/ I*x -I*x\ / -I*x I*x\
|e e | /3 ___\ I*\- e + e /
|---- + -----|*atan\\/ x / - ------------------
\ 2 2 / 2/3 / 2/3\
6*x *\1 + x /
$$\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \operatorname{atan}{\left(\sqrt[3]{x} \right)} - \frac{i \left(e^{i x} - e^{- i x}\right)}{6 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
(exp(i*x)/2 + exp(-i*x)/2)*atan(x^(1/3)) - i*(-exp(-i*x) + exp(i*x))/(6*x^(2/3)*(1 + x^(2/3)))
Parte trigonométrica
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/ 2/x\\ /3 ___\ /x\
|1 - tan |-||*atan\\/ x / 2*tan|-|
\ \2// \2/
------------------------- + -------------------------------
2/x\ 2/3 / 2/3\ / 2/x\\
1 + tan |-| 3*x *\1 + x /*|1 + tan |-||
\2/ \ \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \operatorname{atan}{\left(\sqrt[3]{x} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/x\\ /3 ___\ /x\
|-1 + cot |-||*atan\\/ x / 2*cot|-|
\ \2// \2/
-------------------------- + -------------------------------
2/x\ 2/3 / 2/3\ / 2/x\\
1 + cot |-| 3*x *\1 + x /*|1 + cot |-||
\2/ \ \2//
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \operatorname{atan}{\left(\sqrt[3]{x} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ pi\
cos|x - --|
/3 ___\ \ 2 /
atan\\/ x /*cos(x) + -----------------
2/3 / 2/3\
3*x *\1 + x /
$$\cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
/3 ___\ atan\\/ x / 1 ----------- + ------------------------ /pi \ 2/3 / 2/3\ csc|-- - x| 3*x *\1 + x /*csc(x) \2 /
$$\frac{\operatorname{atan}{\left(\sqrt[3]{x} \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \csc{\left(x \right)}}$$
1 /3 ___\ -----------------*sin(x) + atan\\/ x /*cos(x) 2/3 / 2/3\ 3*x *\1 + x /
$$\frac{1}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)} \sin{\left(x \right)} + \cos{\left(x \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)}$$
/3 ___\
atan\\/ x / 1
----------- + ------------------------
sec(x) 2/3 / 2/3\
3*x *\1 + x /*csc(x)
$$\frac{\operatorname{atan}{\left(\sqrt[3]{x} \right)}}{\sec{\left(x \right)}} + \frac{1}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \csc{\left(x \right)}}$$
/3 ___\ / pi\ sin(x)
atan\\/ x /*sin|x + --| + -----------------
\ 2 / 2/3 / 2/3\
3*x *\1 + x /
$$\sin{\left(x + \frac{\pi}{2} \right)} \operatorname{atan}{\left(\sqrt[3]{x} \right)} + \frac{\sin{\left(x \right)}}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right)}$$
/3 ___\
atan\\/ x / 1
----------- + -----------------------------
sec(x) 2/3 / 2/3\ / pi\
3*x *\1 + x /*sec|x - --|
\ 2 /
$$\frac{\operatorname{atan}{\left(\sqrt[3]{x} \right)}}{\sec{\left(x \right)}} + \frac{1}{3 x^{\frac{2}{3}} \left(x^{\frac{2}{3}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
atan(x^(1/3))/sec(x) + 1/(3*x^(2/3)*(1 + x^(2/3))*sec(x - pi/2))