Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
atan(3*x)^(sqrt(x))*(log(atan(3*x))/(2*sqrt(x))+3*sqrt(x)/((1+9*x^2)*atan(3*x)))
¿Cómo vas a descomponer esta atan(3*x)^(sqrt(x))*(log(atan(3*x))/(2*sqrt(x))+3*sqrt(x)/((1+9*x^2)*atan(3*x))) expresión en fracciones?
Solución
Ha introducido
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___ / ___ \
\/ x |log(atan(3*x)) 3*\/ x |
(atan(3*x)) *|-------------- + --------------------|
| ___ / 2\ |
\ 2*\/ x \1 + 9*x /*atan(3*x)/
$$\left(\frac{3 \sqrt{x}}{\left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}} + \frac{\log{\left(\operatorname{atan}{\left(3 x \right)} \right)}}{2 \sqrt{x}}\right) \operatorname{atan}^{\sqrt{x}}{\left(3 x \right)}$$
atan(3*x)^(sqrt(x))*(log(atan(3*x))/((2*sqrt(x))) + (3*sqrt(x))/(((1 + 9*x^2)*atan(3*x))))
Simplificación general
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___
-1 + \/ x / / 2\ \
(atan(3*x)) *\6*x + \1 + 9*x /*atan(3*x)*log(atan(3*x))/
-----------------------------------------------------------------
___ / 2\
2*\/ x *\1 + 9*x /
$$\frac{\left(6 x + \left(9 x^{2} + 1\right) \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)}\right) \operatorname{atan}^{\sqrt{x} - 1}{\left(3 x \right)}}{2 \sqrt{x} \left(9 x^{2} + 1\right)}$$
atan(3*x)^(-1 + sqrt(x))*(6*x + (1 + 9*x^2)*atan(3*x)*log(atan(3*x)))/(2*sqrt(x)*(1 + 9*x^2))
Respuesta numérica
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atan(3*x)^(x^0.5)*(0.5*x^(-0.5)*log(atan(3*x)) + 3.0*x^0.5/((1.0 + 9.0*x^2)*atan(3*x)))
atan(3*x)^(x^0.5)*(0.5*x^(-0.5)*log(atan(3*x)) + 3.0*x^0.5/((1.0 + 9.0*x^2)*atan(3*x)))
Denominador común
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___ ___ ___
\/ x \/ x 2 \/ x
6*x*(atan(3*x)) + (atan(3*x)) *atan(3*x)*log(atan(3*x)) + 9*x *(atan(3*x)) *atan(3*x)*log(atan(3*x))
-----------------------------------------------------------------------------------------------------------------
___ 5/2
2*\/ x *atan(3*x) + 18*x *atan(3*x)
$$\frac{9 x^{2} \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)} \operatorname{atan}^{\sqrt{x}}{\left(3 x \right)} + 6 x \operatorname{atan}^{\sqrt{x}}{\left(3 x \right)} + \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)} \operatorname{atan}^{\sqrt{x}}{\left(3 x \right)}}{18 x^{\frac{5}{2}} \operatorname{atan}{\left(3 x \right)} + 2 \sqrt{x} \operatorname{atan}{\left(3 x \right)}}$$
(6*x*atan(3*x)^(sqrt(x)) + atan(3*x)^(sqrt(x))*atan(3*x)*log(atan(3*x)) + 9*x^2*atan(3*x)^(sqrt(x))*atan(3*x)*log(atan(3*x)))/(2*sqrt(x)*atan(3*x) + 18*x^(5/2)*atan(3*x))
Combinatoria
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___
\/ x / 2 \
(atan(3*x)) *\6*x + atan(3*x)*log(atan(3*x)) + 9*x *atan(3*x)*log(atan(3*x))/
---------------------------------------------------------------------------------
___ / 2\
2*\/ x *\1 + 9*x /*atan(3*x)
$$\frac{\left(9 x^{2} \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)} + 6 x + \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)}\right) \operatorname{atan}^{\sqrt{x}}{\left(3 x \right)}}{2 \sqrt{x} \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}}$$
atan(3*x)^(sqrt(x))*(6*x + atan(3*x)*log(atan(3*x)) + 9*x^2*atan(3*x)*log(atan(3*x)))/(2*sqrt(x)*(1 + 9*x^2)*atan(3*x))
Unión de expresiones racionales
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___
\/ x / / 2\ \
(atan(3*x)) *\6*x + \1 + 9*x /*atan(3*x)*log(atan(3*x))/
------------------------------------------------------------
___ / 2\
2*\/ x *\1 + 9*x /*atan(3*x)
$$\frac{\left(6 x + \left(9 x^{2} + 1\right) \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)}\right) \operatorname{atan}^{\sqrt{x}}{\left(3 x \right)}}{2 \sqrt{x} \left(9 x^{2} + 1\right) \operatorname{atan}{\left(3 x \right)}}$$
atan(3*x)^(sqrt(x))*(6*x + (1 + 9*x^2)*atan(3*x)*log(atan(3*x)))/(2*sqrt(x)*(1 + 9*x^2)*atan(3*x))
Denominador racional
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___ ___ ___
-1 + \/ x -1 + \/ x 2 -1 + \/ x
6*x*(atan(3*x)) + (atan(3*x)) *atan(3*x)*log(atan(3*x)) + 9*x *(atan(3*x)) *atan(3*x)*log(atan(3*x))
--------------------------------------------------------------------------------------------------------------------------------
___ / 2\
2*\/ x *\1 + 9*x /
$$\frac{9 x^{2} \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)} \operatorname{atan}^{\sqrt{x} - 1}{\left(3 x \right)} + 6 x \operatorname{atan}^{\sqrt{x} - 1}{\left(3 x \right)} + \log{\left(\operatorname{atan}{\left(3 x \right)} \right)} \operatorname{atan}{\left(3 x \right)} \operatorname{atan}^{\sqrt{x} - 1}{\left(3 x \right)}}{2 \sqrt{x} \left(9 x^{2} + 1\right)}$$
(6*x*atan(3*x)^(-1 + sqrt(x)) + atan(3*x)^(-1 + sqrt(x))*atan(3*x)*log(atan(3*x)) + 9*x^2*atan(3*x)^(-1 + sqrt(x))*atan(3*x)*log(atan(3*x)))/(2*sqrt(x)*(1 + 9*x^2))