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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(sqrt(x))/(pi+2)+(x+1)/(2*sqrt(x)*(1+x)*(pi+2))

¿Cómo vas a descomponer esta atan(sqrt(x))/(pi+2)+(x+1)/(2*sqrt(x)*(1+x)*(pi+2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /  ___\                           
atan\\/ x /            x + 1          
----------- + ------------------------
   pi + 2         ___                 
              2*\/ x *(1 + x)*(pi + 2)
atan⁡(x)2+π+x+12x(x+1)(2+π)\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{2 + \pi} + \frac{x + 1}{2 \sqrt{x} \left(x + 1\right) \left(2 + \pi\right)} 
atan(sqrt(x))/(pi + 2) + (x + 1)/((((2*sqrt(x))*(1 + x))*(pi + 2)))
Simplificación general [src] 
1     ___     /  ___\
- + \/ x *atan\\/ x /
2                    
---------------------
      ___            
    \/ x *(2 + pi)
xatan⁡(x)+12x(2+π)\frac{\sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + \frac{1}{2}}{\sqrt{x} \left(2 + \pi\right)} 
(1/2 + sqrt(x)*atan(sqrt(x)))/(sqrt(x)*(2 + pi))
Respuesta numérica [src] 
0.0972461324120857*x^(-0.5) + 0.194492264824171*atan(sqrt(x))
0.0972461324120857*x^(-0.5) + 0.194492264824171*atan(sqrt(x))
Combinatoria [src] 
        ___     /  ___\
1 + 2*\/ x *atan\\/ x /
-----------------------
        ___            
    2*\/ x *(2 + pi)
2xatan⁡(x)+12x(2+π)\frac{2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + 1}{2 \sqrt{x} \left(2 + \pi\right)} 
(1 + 2*sqrt(x)*atan(sqrt(x)))/(2*sqrt(x)*(2 + pi))
Denominador común [src] 
        ___     /  ___\
1 + 2*\/ x *atan\\/ x /
-----------------------
      ___          ___ 
  4*\/ x  + 2*pi*\/ x
2xatan⁡(x)+14x+2πx\frac{2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + 1}{4 \sqrt{x} + 2 \pi \sqrt{x}} 
(1 + 2*sqrt(x)*atan(sqrt(x)))/(4*sqrt(x) + 2*pi*sqrt(x))
Compilar la expresión [src] 
    /  ___\                   
atan\\/ x /          1        
----------- + ----------------
   2 + pi         ___         
              2*\/ x *(2 + pi)
atan⁡(x)2+π+12x(2+π)\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{2 + \pi} + \frac{1}{2 \sqrt{x} \left(2 + \pi\right)} 
atan(sqrt(x))/(2 + pi) + 1/(2*sqrt(x)*(2 + pi))
Abrimos la expresión [src] 
    /  ___\                           
atan\\/ x /            x + 1          
----------- + ------------------------
   pi + 2         ___                 
              2*\/ x *(1 + x)*(pi + 2)
atan⁡(x)2+π+x+12x(2+π)(x+1)\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{2 + \pi} + \frac{x + 1}{2 \sqrt{x} \left(2 + \pi\right) \left(x + 1\right)} 
atan(sqrt(x))/(pi + 2) + (x + 1)/(2*sqrt(x)*(1 + x)*(pi + 2))
Denominador racional [src] 
    ___      3/2        ___       3/2           /  ___\      2     /  ___\              /  ___\         2     /  ___\
2*\/ x  + 2*x    + pi*\/ x  + pi*x    + 4*x*atan\\/ x / + 4*x *atan\\/ x / + 2*pi*x*atan\\/ x / + 2*pi*x *atan\\/ x /
---------------------------------------------------------------------------------------------------------------------
                                                                    2                                                
                                                2*x*(1 + x)*(2 + pi)
2x32+πx32+2x+πx+4x2atan⁡(x)+2πx2atan⁡(x)+4xatan⁡(x)+2πxatan⁡(x)2x(2+π)2(x+1)\frac{2 x^{\frac{3}{2}} + \pi x^{\frac{3}{2}} + 2 \sqrt{x} + \pi \sqrt{x} + 4 x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} + 2 \pi x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} + 4 x \operatorname{atan}{\left(\sqrt{x} \right)} + 2 \pi x \operatorname{atan}{\left(\sqrt{x} \right)}}{2 x \left(2 + \pi\right)^{2} \left(x + 1\right)} 
(2*sqrt(x) + 2*x^(3/2) + pi*sqrt(x) + pi*x^(3/2) + 4*x*atan(sqrt(x)) + 4*x^2*atan(sqrt(x)) + 2*pi*x*atan(sqrt(x)) + 2*pi*x^2*atan(sqrt(x)))/(2*x*(1 + x)*(2 + pi)^2)
Parte trigonométrica [src] 
    /  ___\                   
atan\\/ x /          1        
----------- + ----------------
   2 + pi         ___         
              2*\/ x *(2 + pi)
atan⁡(x)2+π+12x(2+π)\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{2 + \pi} + \frac{1}{2 \sqrt{x} \left(2 + \pi\right)} 
atan(sqrt(x))/(2 + pi) + 1/(2*sqrt(x)*(2 + pi))
Potencias [src] 
    /  ___\                   
atan\\/ x /          1        
----------- + ----------------
   2 + pi         ___         
              2*\/ x *(2 + pi)
atan⁡(x)2+π+12x(2+π)\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{2 + \pi} + \frac{1}{2 \sqrt{x} \left(2 + \pi\right)} 
    /  ___\                           
atan\\/ x /            1 + x          
----------- + ------------------------
   2 + pi       ___                   
              \/ x *(2 + pi)*(2 + 2*x)
atan⁡(x)2+π+x+1x(2+π)(2x+2)\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{2 + \pi} + \frac{x + 1}{\sqrt{x} \left(2 + \pi\right) \left(2 x + 2\right)} 
atan(sqrt(x))/(2 + pi) + (1 + x)/(sqrt(x)*(2 + pi)*(2 + 2*x))
Unión de expresiones racionales [src] 
        ___     /  ___\
1 + 2*\/ x *atan\\/ x /
-----------------------
        ___            
    2*\/ x *(2 + pi)
2xatan⁡(x)+12x(2+π)\frac{2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + 1}{2 \sqrt{x} \left(2 + \pi\right)} 
(1 + 2*sqrt(x)*atan(sqrt(x)))/(2*sqrt(x)*(2 + pi))
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