Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(pi+2)/atan(sqrt(x))-(pi+2)*(x+1)/(2*sqrt(x)*(1+x)*atan(sqrt(x))^2)
¿Cómo vas a descomponer esta (pi+2)/atan(sqrt(x))-(pi+2)*(x+1)/(2*sqrt(x)*(1+x)*atan(sqrt(x))^2) expresión en fracciones?
Solución
Ha introducido
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pi + 2 (pi + 2)*(x + 1)
----------- - ----------------------------
/ ___\ ___ 2/ ___\
atan\\/ x / 2*\/ x *(1 + x)*atan \\/ x /
$$\frac{2 + \pi}{\operatorname{atan}{\left(\sqrt{x} \right)}} - \frac{\left(2 + \pi\right) \left(x + 1\right)}{2 \sqrt{x} \left(x + 1\right) \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(pi + 2)/atan(sqrt(x)) - (pi + 2)*(x + 1)/(((2*sqrt(x))*(1 + x))*atan(sqrt(x))^2)
Simplificación general
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pi ___ / ___\
-1 - -- + \/ x *(2 + pi)*atan\\/ x /
2
------------------------------------
___ 2/ ___\
\/ x *atan \\/ x /
$$\frac{\sqrt{x} \left(2 + \pi\right) \operatorname{atan}{\left(\sqrt{x} \right)} - \frac{\pi}{2} - 1}{\sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(-1 - pi/2 + sqrt(x)*(2 + pi)*atan(sqrt(x)))/(sqrt(x)*atan(sqrt(x))^2)
Respuesta numérica
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5.14159265358979/atan(sqrt(x)) - 0.5*x^(-0.5)*(5.14159265358979 + 5.14159265358979*x)/((1.0 + x)*atan(sqrt(x))^2)
5.14159265358979/atan(sqrt(x)) - 0.5*x^(-0.5)*(5.14159265358979 + 5.14159265358979*x)/((1.0 + x)*atan(sqrt(x))^2)
Parte trigonométrica
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2 + pi 2 + pi
----------- - --------------------
/ ___\ ___ 2/ ___\
atan\\/ x / 2*\/ x *atan \\/ x /
$$\frac{2 + \pi}{\operatorname{atan}{\left(\sqrt{x} \right)}} - \frac{2 + \pi}{2 \sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(2 + pi)/atan(sqrt(x)) - (2 + pi)/(2*sqrt(x)*atan(sqrt(x))^2)
Denominador común
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___ / ___\ ___ / ___\
-2 - pi + 4*\/ x *atan\\/ x / + 2*pi*\/ x *atan\\/ x /
------------------------------------------------------
___ 2/ ___\
2*\/ x *atan \\/ x /
$$\frac{4 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + 2 \pi \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} - \pi - 2}{2 \sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(-2 - pi + 4*sqrt(x)*atan(sqrt(x)) + 2*pi*sqrt(x)*atan(sqrt(x)))/(2*sqrt(x)*atan(sqrt(x))^2)
Combinatoria
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/ ___ / ___\\
\-1 + 2*\/ x *atan\\/ x //*(2 + pi)
-----------------------------------
___ 2/ ___\
2*\/ x *atan \\/ x /
$$\frac{\left(2 + \pi\right) \left(2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} - 1\right)}{2 \sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(-1 + 2*sqrt(x)*atan(sqrt(x)))*(2 + pi)/(2*sqrt(x)*atan(sqrt(x))^2)
Compilar la expresión
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2 + pi 2 + pi
----------- - --------------------
/ ___\ ___ 2/ ___\
atan\\/ x / 2*\/ x *atan \\/ x /
$$\frac{2 + \pi}{\operatorname{atan}{\left(\sqrt{x} \right)}} - \frac{2 + \pi}{2 \sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(2 + pi)/atan(sqrt(x)) - (2 + pi)/(2*sqrt(x)*atan(sqrt(x))^2)
Denominador racional
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___ / ___\ 3/2 / ___\ 2/ ___\ 2 2/ ___\ ___ / ___\ 3/2 / ___\ 2/ ___\ 2 2/ ___\
- 2*\/ x *atan\\/ x / - 2*x *atan\\/ x / + 4*x*atan \\/ x / + 4*x *atan \\/ x / - pi*\/ x *atan\\/ x / - pi*x *atan\\/ x / + 2*pi*x*atan \\/ x / + 2*pi*x *atan \\/ x /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3/ ___\
2*x*(1 + x)*atan \\/ x /
$$\frac{- \pi x^{\frac{3}{2}} \operatorname{atan}{\left(\sqrt{x} \right)} - 2 x^{\frac{3}{2}} \operatorname{atan}{\left(\sqrt{x} \right)} - \pi \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} - 2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + 4 x^{2} \operatorname{atan}^{2}{\left(\sqrt{x} \right)} + 2 \pi x^{2} \operatorname{atan}^{2}{\left(\sqrt{x} \right)} + 4 x \operatorname{atan}^{2}{\left(\sqrt{x} \right)} + 2 \pi x \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}{2 x \left(x + 1\right) \operatorname{atan}^{3}{\left(\sqrt{x} \right)}}$$
(-2*sqrt(x)*atan(sqrt(x)) - 2*x^(3/2)*atan(sqrt(x)) + 4*x*atan(sqrt(x))^2 + 4*x^2*atan(sqrt(x))^2 - pi*sqrt(x)*atan(sqrt(x)) - pi*x^(3/2)*atan(sqrt(x)) + 2*pi*x*atan(sqrt(x))^2 + 2*pi*x^2*atan(sqrt(x))^2)/(2*x*(1 + x)*atan(sqrt(x))^3)
Potencias
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2 + pi 2 + pi
----------- - --------------------
/ ___\ ___ 2/ ___\
atan\\/ x / 2*\/ x *atan \\/ x /
$$\frac{2 + \pi}{\operatorname{atan}{\left(\sqrt{x} \right)}} - \frac{2 + \pi}{2 \sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
2 + pi (1 + x)*(2 + pi)
----------- - ----------------------------
/ ___\ ___ 2/ ___\
atan\\/ x / \/ x *(2 + 2*x)*atan \\/ x /
$$\frac{2 + \pi}{\operatorname{atan}{\left(\sqrt{x} \right)}} - \frac{\left(2 + \pi\right) \left(x + 1\right)}{\sqrt{x} \left(2 x + 2\right) \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(2 + pi)/atan(sqrt(x)) - (1 + x)*(2 + pi)/(sqrt(x)*(2 + 2*x)*atan(sqrt(x))^2)
Unión de expresiones racionales
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/ ___ / ___\\
\-1 + 2*\/ x *atan\\/ x //*(2 + pi)
-----------------------------------
___ 2/ ___\
2*\/ x *atan \\/ x /
$$\frac{\left(2 + \pi\right) \left(2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} - 1\right)}{2 \sqrt{x} \operatorname{atan}^{2}{\left(\sqrt{x} \right)}}$$
(-1 + 2*sqrt(x)*atan(sqrt(x)))*(2 + pi)/(2*sqrt(x)*atan(sqrt(x))^2)