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Simplificación de expresiones/ Descomposición en fracciones simples/ acos(x)/x+1/2*log(1-sqrt(1-x^2))/(1+sqrt(1-x^2))

¿Cómo vas a descomponer esta acos(x)/x+1/2*log(1-sqrt(1-x^2))/(1+sqrt(1-x^2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
          /   /       ________\\
          |   |      /      2 ||
          |log\1 - \/  1 - x  /|
          |--------------------|
acos(x)   \         2          /
------- + ----------------------
   x                ________    
                   /      2     
             1 + \/  1 - x
12log⁡(1−1−x2)1−x2+1+acos⁡(x)x\frac{\frac{1}{2} \log{\left(1 - \sqrt{1 - x^{2}} \right)}}{\sqrt{1 - x^{2}} + 1} + \frac{\operatorname{acos}{\left(x \right)}}{x} 
acos(x)/x + (log(1 - sqrt(1 - x^2))/2)/(1 + sqrt(1 - x^2))
Simplificación general [src] 
                                 /       ________\
/       ________\                |      /      2 |
|      /      2 |           x*log\1 - \/  1 - x  /
\1 + \/  1 - x  /*acos(x) + ----------------------
                                      2           
--------------------------------------------------
                 /       ________\                
                 |      /      2 |                
               x*\1 + \/  1 - x  /
xlog⁡(1−1−x2)2+(1−x2+1)acos⁡(x)x(1−x2+1)\frac{\frac{x \log{\left(1 - \sqrt{1 - x^{2}} \right)}}{2} + \left(\sqrt{1 - x^{2}} + 1\right) \operatorname{acos}{\left(x \right)}}{x \left(\sqrt{1 - x^{2}} + 1\right)} 
((1 + sqrt(1 - x^2))*acos(x) + x*log(1 - sqrt(1 - x^2))/2)/(x*(1 + sqrt(1 - x^2)))
Respuesta numérica [src] 
acos(x)/x + 0.5*log(1 - sqrt(1 - x^2))/(1.0 + (1.0 - x^2)^0.5)
acos(x)/x + 0.5*log(1 - sqrt(1 - x^2))/(1.0 + (1.0 - x^2)^0.5)
Potencias [src] 
             /       ________\
             |      /      2 |
acos(x)   log\1 - \/  1 - x  /
------- + --------------------
   x        /       ________\ 
            |      /      2 | 
          2*\1 + \/  1 - x  /
log⁡(1−1−x2)2(1−x2+1)+acos⁡(x)x\frac{\log{\left(1 - \sqrt{1 - x^{2}} \right)}}{2 \left(\sqrt{1 - x^{2}} + 1\right)} + \frac{\operatorname{acos}{\left(x \right)}}{x} 
acos(x)/x + log(1 - sqrt(1 - x^2))/(2*(1 + sqrt(1 - x^2)))
Denominador común [src] 
                 /       ________\        ________        
                 |      /      2 |       /      2         
2*acos(x) + x*log\1 - \/  1 - x  / + 2*\/  1 - x  *acos(x)
----------------------------------------------------------
                               ________                   
                              /      2                    
                  2*x + 2*x*\/  1 - x
xlog⁡(1−1−x2)+21−x2acos⁡(x)+2acos⁡(x)2x1−x2+2x\frac{x \log{\left(1 - \sqrt{1 - x^{2}} \right)} + 2 \sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} + 2 \operatorname{acos}{\left(x \right)}}{2 x \sqrt{1 - x^{2}} + 2 x} 
(2*acos(x) + x*log(1 - sqrt(1 - x^2)) + 2*sqrt(1 - x^2)*acos(x))/(2*x + 2*x*sqrt(1 - x^2))
Parte trigonométrica [src] 
             /       ________\
             |      /      2 |
acos(x)   log\1 - \/  1 - x  /
------- + --------------------
   x        /       ________\ 
            |      /      2 | 
          2*\1 + \/  1 - x  /
log⁡(1−1−x2)2(1−x2+1)+acos⁡(x)x\frac{\log{\left(1 - \sqrt{1 - x^{2}} \right)}}{2 \left(\sqrt{1 - x^{2}} + 1\right)} + \frac{\operatorname{acos}{\left(x \right)}}{x} 
acos(x)/x + log(1 - sqrt(1 - x^2))/(2*(1 + sqrt(1 - x^2)))
Denominador racional [src] 
       /       ________\                         ________    /       ________\
       |      /      2 |      2                 /      2     |      /      2 |
2*x*log\1 - \/  1 - x  / + 4*x *acos(x) - 2*x*\/  1 - x  *log\1 - \/  1 - x  /
------------------------------------------------------------------------------
                                        3                                     
                                     4*x
4x2acos⁡(x)−2x1−x2log⁡(1−1−x2)+2xlog⁡(1−1−x2)4x3\frac{4 x^{2} \operatorname{acos}{\left(x \right)} - 2 x \sqrt{1 - x^{2}} \log{\left(1 - \sqrt{1 - x^{2}} \right)} + 2 x \log{\left(1 - \sqrt{1 - x^{2}} \right)}}{4 x^{3}} 
(2*x*log(1 - sqrt(1 - x^2)) + 4*x^2*acos(x) - 2*x*sqrt(1 - x^2)*log(1 - sqrt(1 - x^2)))/(4*x^3)
Unión de expresiones racionales [src] 
     /       ________\     /       ________\        
     |      /      2 |     |      /      2 |        
x*log\1 - \/  1 - x  / + 2*\1 + \/  1 - x  /*acos(x)
----------------------------------------------------
                   /       ________\                
                   |      /      2 |                
               2*x*\1 + \/  1 - x  /
xlog⁡(1−1−x2)+2(1−x2+1)acos⁡(x)2x(1−x2+1)\frac{x \log{\left(1 - \sqrt{1 - x^{2}} \right)} + 2 \left(\sqrt{1 - x^{2}} + 1\right) \operatorname{acos}{\left(x \right)}}{2 x \left(\sqrt{1 - x^{2}} + 1\right)} 
(x*log(1 - sqrt(1 - x^2)) + 2*(1 + sqrt(1 - x^2))*acos(x))/(2*x*(1 + sqrt(1 - x^2)))
Compilar la expresión [src] 
             /       ________\
             |      /      2 |
acos(x)   log\1 - \/  1 - x  /
------- + --------------------
   x        /       ________\ 
            |      /      2 | 
          2*\1 + \/  1 - x  /
log⁡(1−1−x2)2(1−x2+1)+acos⁡(x)x\frac{\log{\left(1 - \sqrt{1 - x^{2}} \right)}}{2 \left(\sqrt{1 - x^{2}} + 1\right)} + \frac{\operatorname{acos}{\left(x \right)}}{x} 
acos(x)/x + log(1 - sqrt(1 - x^2))/(2*(1 + sqrt(1 - x^2)))
Combinatoria [src] 
                 /       ________\        ________        
                 |      /      2 |       /      2         
2*acos(x) + x*log\1 - \/  1 - x  / + 2*\/  1 - x  *acos(x)
----------------------------------------------------------
                      /       ________\                   
                      |      /      2 |                   
                  2*x*\1 + \/  1 - x  /
xlog⁡(1−1−x2)+21−x2acos⁡(x)+2acos⁡(x)2x(1−x2+1)\frac{x \log{\left(1 - \sqrt{1 - x^{2}} \right)} + 2 \sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} + 2 \operatorname{acos}{\left(x \right)}}{2 x \left(\sqrt{1 - x^{2}} + 1\right)} 
(2*acos(x) + x*log(1 - sqrt(1 - x^2)) + 2*sqrt(1 - x^2)*acos(x))/(2*x*(1 + sqrt(1 - x^2)))
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