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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
 cos(x)          sin(x)       
------- + --------------------
acos(x)      ________         
            /      2      2   
          \/  1 - x  *acos (x)
cos⁡(x)acos⁡(x)+sin⁡(x)1−x2acos⁡2(x)\frac{\cos{\left(x \right)}}{\operatorname{acos}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)}} 
cos(x)/acos(x) + sin(x)/((sqrt(1 - x^2)*acos(x)^2))
Respuesta numérica [src] 
cos(x)/acos(x) + (1.0 - x^2)^(-0.5)*sin(x)/acos(x)^2
cos(x)/acos(x) + (1.0 - x^2)^(-0.5)*sin(x)/acos(x)^2
Combinatoria [src] 
   ________                        
  /      2                         
\/  1 - x  *acos(x)*cos(x) + sin(x)
-----------------------------------
     ___________________     2     
   \/ -(1 + x)*(-1 + x) *acos (x)
1−x2cos⁡(x)acos⁡(x)+sin⁡(x)−(x−1)(x+1)acos⁡2(x)\frac{\sqrt{1 - x^{2}} \cos{\left(x \right)} \operatorname{acos}{\left(x \right)} + \sin{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \operatorname{acos}^{2}{\left(x \right)}} 
(sqrt(1 - x^2)*acos(x)*cos(x) + sin(x))/(sqrt(-(1 + x)*(-1 + x))*acos(x)^2)
Denominador común [src] 
   ________                        
  /      2                         
\/  1 - x  *acos(x)*cos(x) + sin(x)
-----------------------------------
           ________                
          /      2      2          
        \/  1 - x  *acos (x)
1−x2cos⁡(x)acos⁡(x)+sin⁡(x)1−x2acos⁡2(x)\frac{\sqrt{1 - x^{2}} \cos{\left(x \right)} \operatorname{acos}{\left(x \right)} + \sin{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)}} 
(sqrt(1 - x^2)*acos(x)*cos(x) + sin(x))/(sqrt(1 - x^2)*acos(x)^2)
Denominador racional [src] 
                                            ________               
      2              2     2               /      2                
- acos (x)*cos(x) + x *acos (x)*cos(x) - \/  1 - x  *acos(x)*sin(x)
-------------------------------------------------------------------
                         /      2\     3                           
                         \-1 + x /*acos (x)
x2cos⁡(x)acos⁡2(x)−1−x2sin⁡(x)acos⁡(x)−cos⁡(x)acos⁡2(x)(x2−1)acos⁡3(x)\frac{x^{2} \cos{\left(x \right)} \operatorname{acos}^{2}{\left(x \right)} - \sqrt{1 - x^{2}} \sin{\left(x \right)} \operatorname{acos}{\left(x \right)} - \cos{\left(x \right)} \operatorname{acos}^{2}{\left(x \right)}}{\left(x^{2} - 1\right) \operatorname{acos}^{3}{\left(x \right)}} 
(-acos(x)^2*cos(x) + x^2*acos(x)^2*cos(x) - sqrt(1 - x^2)*acos(x)*sin(x))/((-1 + x^2)*acos(x)^3)
Parte trigonométrica [src] 
      1                         1                
-------------- + --------------------------------
acos(x)*sec(x)      ________                     
                   /      2      2       /    pi\
                 \/  1 - x  *acos (x)*sec|x - --|
                                         \    2 /
1acos⁡(x)sec⁡(x)+11−x2acos⁡2(x)sec⁡(x−π2)\frac{1}{\operatorname{acos}{\left(x \right)} \sec{\left(x \right)}} + \frac{1}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} 
       sin(x)            /    pi\
------------------- + sin|x + --|
   ________              \    2 /
  /      2                       
\/  1 - x  *acos(x)              
---------------------------------
             acos(x)
sin⁡(x+π2)+sin⁡(x)1−x2acos⁡(x)acos⁡(x)\frac{\sin{\left(x + \frac{\pi}{2} \right)} + \frac{\sin{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}}{\operatorname{acos}{\left(x \right)}} 
  1                     1               
------ + -------------------------------
sec(x)      ________                    
           /      2             /    pi\
         \/  1 - x  *acos(x)*sec|x - --|
                                \    2 /
----------------------------------------
                acos(x)
1sec⁡(x)+11−x2acos⁡(x)sec⁡(x−π2)acos⁡(x)\frac{\frac{1}{\sec{\left(x \right)}} + \frac{1}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}}{\operatorname{acos}{\left(x \right)}} 
       /    pi\             
    cos|x - --|             
       \    2 /             
------------------- + cos(x)
   ________                 
  /      2                  
\/  1 - x  *acos(x)         
----------------------------
          acos(x)
cos⁡(x)+cos⁡(x−π2)1−x2acos⁡(x)acos⁡(x)\frac{\cos{\left(x \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}}{\operatorname{acos}{\left(x \right)}} 
            2/x\                          /x\             
     1 - tan |-|                     2*tan|-|             
             \2/                          \2/             
--------------------- + ----------------------------------
/       2/x\\                            ________         
|1 + tan |-||*acos(x)   /       2/x\\   /      2      2   
\        \2//           |1 + tan |-||*\/  1 - x  *acos (x)
                        \        \2//
1−tan⁡2(x2)(tan⁡2(x2)+1)acos⁡(x)+2tan⁡(x2)1−x2(tan⁡2(x2)+1)acos⁡2(x)\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{acos}{\left(x \right)}} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\sqrt{1 - x^{2}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{acos}^{2}{\left(x \right)}} 
        2/x\                     /x\            
-1 + cot |-|                2*cot|-|            
         \2/                     \2/            
------------ + ---------------------------------
       2/x\                     ________        
1 + cot |-|    /       2/x\\   /      2         
        \2/    |1 + cot |-||*\/  1 - x  *acos(x)
               \        \2//                    
------------------------------------------------
                    acos(x)
cot⁡2(x2)−1cot⁡2(x2)+1+2cot⁡(x2)1−x2(cot⁡2(x2)+1)acos⁡(x)acos⁡(x)\frac{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\sqrt{1 - x^{2}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{acos}{\left(x \right)}}}{\operatorname{acos}{\left(x \right)}} 
         1                         1             
------------------- + ---------------------------
           /pi    \      ________                
acos(x)*csc|-- - x|     /      2      2          
           \2     /   \/  1 - x  *acos (x)*csc(x)
1acos⁡(x)csc⁡(−x+π2)+11−x2acos⁡2(x)csc⁡(x)\frac{1}{\operatorname{acos}{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)} \csc{\left(x \right)}} 
   /    pi\                       
sin|x + --|                       
   \    2 /          sin(x)       
----------- + --------------------
  acos(x)        ________         
                /      2      2   
              \/  1 - x  *acos (x)
sin⁡(x+π2)acos⁡(x)+sin⁡(x)1−x2acos⁡2(x)\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\operatorname{acos}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)}} 
                 /    pi\     
              cos|x - --|     
 cos(x)          \    2 /     
------- + --------------------
acos(x)      ________         
            /      2      2   
          \/  1 - x  *acos (x)
cos⁡(x)acos⁡(x)+cos⁡(x−π2)1−x2acos⁡2(x)\frac{\cos{\left(x \right)}}{\operatorname{acos}{\left(x \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)}} 
      1                       1             
-------------- + ---------------------------
acos(x)*sec(x)      ________                
                   /      2      2          
                 \/  1 - x  *acos (x)*csc(x)
1acos⁡(x)sec⁡(x)+11−x2acos⁡2(x)csc⁡(x)\frac{1}{\operatorname{acos}{\left(x \right)} \sec{\left(x \right)}} + \frac{1}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)} \csc{\left(x \right)}} 
     1                    1             
----------- + --------------------------
   /pi    \      ________               
csc|-- - x|     /      2                
   \2     /   \/  1 - x  *acos(x)*csc(x)
----------------------------------------
                acos(x)
1csc⁡(−x+π2)+11−x2acos⁡(x)csc⁡(x)acos⁡(x)\frac{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \csc{\left(x \right)}}}{\operatorname{acos}{\left(x \right)}} 
             2/x\                         /x\             
     -1 + cot |-|                    2*cot|-|             
              \2/                         \2/             
--------------------- + ----------------------------------
/       2/x\\                            ________         
|1 + cot |-||*acos(x)   /       2/x\\   /      2      2   
\        \2//           |1 + cot |-||*\/  1 - x  *acos (x)
                        \        \2//
cot⁡2(x2)−1(cot⁡2(x2)+1)acos⁡(x)+2cot⁡(x2)1−x2(cot⁡2(x2)+1)acos⁡2(x)\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{acos}{\left(x \right)}} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\sqrt{1 - x^{2}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{acos}^{2}{\left(x \right)}} 
       sin(x)               
------------------- + cos(x)
   ________                 
  /      2                  
\/  1 - x  *acos(x)         
----------------------------
          acos(x)
cos⁡(x)+sin⁡(x)1−x2acos⁡(x)acos⁡(x)\frac{\cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}}{\operatorname{acos}{\left(x \right)}} 
       2/x\                     /x\            
1 - tan |-|                2*tan|-|            
        \2/                     \2/            
----------- + ---------------------------------
       2/x\                    ________        
1 + tan |-|   /       2/x\\   /      2         
        \2/   |1 + tan |-||*\/  1 - x  *acos(x)
              \        \2//                    
-----------------------------------------------
                    acos(x)
1−tan⁡2(x2)tan⁡2(x2)+1+2tan⁡(x2)1−x2(tan⁡2(x2)+1)acos⁡(x)acos⁡(x)\frac{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\sqrt{1 - x^{2}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{acos}{\left(x \right)}}}{\operatorname{acos}{\left(x \right)}} 
((1 - tan(x/2)^2)/(1 + tan(x/2)^2) + 2*tan(x/2)/((1 + tan(x/2)^2)*sqrt(1 - x^2)*acos(x)))/acos(x)
Unión de expresiones racionales [src] 
   ________                        
  /      2                         
\/  1 - x  *acos(x)*cos(x) + sin(x)
-----------------------------------
           ________                
          /      2      2          
        \/  1 - x  *acos (x)
1−x2cos⁡(x)acos⁡(x)+sin⁡(x)1−x2acos⁡2(x)\frac{\sqrt{1 - x^{2}} \cos{\left(x \right)} \operatorname{acos}{\left(x \right)} + \sin{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)}} 
(sqrt(1 - x^2)*acos(x)*cos(x) + sin(x))/(sqrt(1 - x^2)*acos(x)^2)
Potencias [src] 
 I*x    -I*x                         
e      e                             
---- + -----       /   -I*x    I*x\  
 2       2       I*\- e     + e   /  
------------ - ----------------------
  acos(x)           ________         
                   /      2      2   
               2*\/  1 - x  *acos (x)
eix2+e−ix2acos⁡(x)−i(eix−e−ix)21−x2acos⁡2(x)\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\operatorname{acos}{\left(x \right)}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \sqrt{1 - x^{2}} \operatorname{acos}^{2}{\left(x \right)}} 
(exp(i*x)/2 + exp(-i*x)/2)/acos(x) - i*(-exp(-i*x) + exp(i*x))/(2*sqrt(1 - x^2)*acos(x)^2)
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