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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
         t*sin(x)  
cos(x) + --------*x
                2  
             t*x   
         1 - ----  
              2
xtsin⁡(x)−tx22+1+cos⁡(x)x \frac{t \sin{\left(x \right)}}{- \frac{t x^{2}}{2} + 1} + \cos{\left(x \right)} 
cos(x) + ((t*sin(x))/(1 - t*x^2/2))*x
Simplificación general [src] 
/        2\                      
\-2 + t*x /*cos(x) - 2*t*x*sin(x)
---------------------------------
                    2            
            -2 + t*x
−2txsin⁡(x)+(tx2−2)cos⁡(x)tx2−2\frac{- 2 t x \sin{\left(x \right)} + \left(t x^{2} - 2\right) \cos{\left(x \right)}}{t x^{2} - 2} 
((-2 + t*x^2)*cos(x) - 2*t*x*sin(x))/(-2 + t*x^2)
Respuesta numérica [src] 
t*x*sin(x)/(1.0 - 0.5*t*x^2) + cos(x)
t*x*sin(x)/(1.0 - 0.5*t*x^2) + cos(x)
Denominador racional [src] 
/       2\                      
\2 - t*x /*cos(x) + 2*t*x*sin(x)
--------------------------------
                   2            
            2 - t*x
2txsin⁡(x)+(−tx2+2)cos⁡(x)−tx2+2\frac{2 t x \sin{\left(x \right)} + \left(- t x^{2} + 2\right) \cos{\left(x \right)}}{- t x^{2} + 2} 
((2 - t*x^2)*cos(x) + 2*t*x*sin(x))/(2 - t*x^2)
Combinatoria [src] 
               2                      
-2*cos(x) + t*x *cos(x) - 2*t*x*sin(x)
--------------------------------------
                      2               
              -2 + t*x
tx2cos⁡(x)−2txsin⁡(x)−2cos⁡(x)tx2−2\frac{t x^{2} \cos{\left(x \right)} - 2 t x \sin{\left(x \right)} - 2 \cos{\left(x \right)}}{t x^{2} - 2} 
(-2*cos(x) + t*x^2*cos(x) - 2*t*x*sin(x))/(-2 + t*x^2)
Potencias [src] 
t*x*sin(x)         
---------- + cos(x)
        2          
     t*x           
 1 - ----          
      2
txsin⁡(x)−tx22+1+cos⁡(x)\frac{t x \sin{\left(x \right)}}{- \frac{t x^{2}}{2} + 1} + \cos{\left(x \right)} 
 I*x    -I*x         /   -I*x    I*x\
e      e       I*t*x*\- e     + e   /
---- + ----- - ----------------------
 2       2            /       2\     
                      |    t*x |     
                    2*|1 - ----|     
                      \     2  /
−itx(eix−e−ix)2(−tx22+1)+eix2+e−ix2- \frac{i t x \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{t x^{2}}{2} + 1\right)} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2} 
exp(i*x)/2 + exp(-i*x)/2 - i*t*x*(-exp(-i*x) + exp(i*x))/(2*(1 - t*x^2/2))
Unión de expresiones racionales [src] 
/       2\                      
\2 - t*x /*cos(x) + 2*t*x*sin(x)
--------------------------------
                   2            
            2 - t*x
2txsin⁡(x)+(−tx2+2)cos⁡(x)−tx2+2\frac{2 t x \sin{\left(x \right)} + \left(- t x^{2} + 2\right) \cos{\left(x \right)}}{- t x^{2} + 2} 
((2 - t*x^2)*cos(x) + 2*t*x*sin(x))/(2 - t*x^2)
Denominador común [src] 
  2*t*x*sin(x)         
- ------------ + cos(x)
           2           
   -2 + t*x
−2txsin⁡(x)tx2−2+cos⁡(x)- \frac{2 t x \sin{\left(x \right)}}{t x^{2} - 2} + \cos{\left(x \right)} 
-2*t*x*sin(x)/(-2 + t*x^2) + cos(x)
Compilar la expresión [src] 
t*x*sin(x)         
---------- + cos(x)
        2          
     t*x           
 1 - ----          
      2
txsin⁡(x)−tx22+1+cos⁡(x)\frac{t x \sin{\left(x \right)}}{- \frac{t x^{2}}{2} + 1} + \cos{\left(x \right)} 
t*x*sin(x)/(1 - t*x^2/2) + cos(x)
Parte trigonométrica [src] 
        2/x\                  /x\      
-1 + cot |-|         2*t*x*cot|-|      
         \2/                  \2/      
------------ + ------------------------
       2/x\                  /       2\
1 + cot |-|    /       2/x\\ |    t*x |
        \2/    |1 + cot |-||*|1 - ----|
               \        \2// \     2  /
2txcot⁡(x2)(−tx22+1)(cot⁡2(x2)+1)+cot⁡2(x2)−1cot⁡2(x2)+1\frac{2 t x \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{t x^{2}}{2} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} 
t*x*sin(x)      /    pi\
---------- + sin|x + --|
        2       \    2 /
     t*x                
 1 - ----               
      2
txsin⁡(x)−tx22+1+sin⁡(x+π2)\frac{t x \sin{\left(x \right)}}{- \frac{t x^{2}}{2} + 1} + \sin{\left(x + \frac{\pi}{2} \right)} 
       2/x\                  /x\      
1 - tan |-|         2*t*x*tan|-|      
        \2/                  \2/      
----------- + ------------------------
       2/x\                 /       2\
1 + tan |-|   /       2/x\\ |    t*x |
        \2/   |1 + tan |-||*|1 - ----|
              \        \2// \     2  /
2txtan⁡(x2)(−tx22+1)(tan⁡2(x2)+1)+1−tan⁡2(x2)tan⁡2(x2)+1\frac{2 t x \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{t x^{2}}{2} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} 
       /    pi\         
t*x*cos|x - --|         
       \    2 /         
--------------- + cos(x)
           2            
        t*x             
    1 - ----            
         2
txcos⁡(x−π2)−tx22+1+cos⁡(x)\frac{t x \cos{\left(x - \frac{\pi}{2} \right)}}{- \frac{t x^{2}}{2} + 1} + \cos{\left(x \right)} 
  1             t*x       
------ + -----------------
sec(x)   /       2\       
         |    t*x |       
         |1 - ----|*csc(x)
         \     2  /
tx(−tx22+1)csc⁡(x)+1sec⁡(x)\frac{t x}{\left(- \frac{t x^{2}}{2} + 1\right) \csc{\left(x \right)}} + \frac{1}{\sec{\left(x \right)}} 
     1               t*x       
----------- + -----------------
   /pi    \   /       2\       
csc|-- - x|   |    t*x |       
   \2     /   |1 - ----|*csc(x)
              \     2  /
tx(−tx22+1)csc⁡(x)+1csc⁡(−x+π2)\frac{t x}{\left(- \frac{t x^{2}}{2} + 1\right) \csc{\left(x \right)}} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} 
t*x*sin(x)         
---------- + cos(x)
        2          
     t*x           
 1 - ----          
      2
txsin⁡(x)−tx22+1+cos⁡(x)\frac{t x \sin{\left(x \right)}}{- \frac{t x^{2}}{2} + 1} + \cos{\left(x \right)} 
  1               t*x          
------ + ----------------------
sec(x)   /       2\            
         |    t*x |    /    pi\
         |1 - ----|*sec|x - --|
         \     2  /    \    2 /
tx(−tx22+1)sec⁡(x−π2)+1sec⁡(x)\frac{t x}{\left(- \frac{t x^{2}}{2} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}} 
1/sec(x) + t*x/((1 - t*x^2/2)*sec(x - pi/2))
Abrimos la expresión [src] 
t*x*sin(x)         
---------- + cos(x)
        2          
     t*x           
 1 - ----          
      2
txsin⁡(x)−tx22+1+cos⁡(x)\frac{t x \sin{\left(x \right)}}{- \frac{t x^{2}}{2} + 1} + \cos{\left(x \right)} 
t*x*sin(x)/(1 - t*x^2/2) + cos(x)
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