Sr Examen

Otras calculadoras

Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x)/(3+cos(x))+(3-cos(x))*sin(x)/(3+cos(x))^2

¿Cómo vas a descomponer esta sin(x)/(3+cos(x))+(3-cos(x))*sin(x)/(3+cos(x))^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(x)     (3 - cos(x))*sin(x)
---------- + -------------------
3 + cos(x)                  2   
                (3 + cos(x))
(3−cos⁡(x))sin⁡(x)(cos⁡(x)+3)2+sin⁡(x)cos⁡(x)+3\frac{\left(3 - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 3} 
sin(x)/(3 + cos(x)) + ((3 - cos(x))*sin(x))/(3 + cos(x))^2
Simplificación general [src] 
   6*sin(x)  
-------------
            2
(3 + cos(x))
6sin⁡(x)(cos⁡(x)+3)2\frac{6 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} 
6*sin(x)/(3 + cos(x))^2
Respuesta numérica [src] 
sin(x)/(3.0 + cos(x)) + 0.111111111111111*(3.0 - cos(x))*sin(x)/(1 + 0.333333333333333*cos(x))^2
sin(x)/(3.0 + cos(x)) + 0.111111111111111*(3.0 - cos(x))*sin(x)/(1 + 0.333333333333333*cos(x))^2
Combinatoria [src] 
   6*sin(x)  
-------------
            2
(3 + cos(x))
6sin⁡(x)(cos⁡(x)+3)2\frac{6 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} 
6*sin(x)/(3 + cos(x))^2
Potencias [src] 
                                            /     I*x    -I*x\
                           /   -I*x    I*x\ |    e      e    |
     /   -I*x    I*x\    I*\- e     + e   /*|3 - ---- - -----|
   I*\- e     + e   /                       \     2       2  /
- -------------------- - -------------------------------------
    /     I*x    -I*x\                               2        
    |    e      e    |             /     I*x    -I*x\         
  2*|3 + ---- + -----|             |    e      e    |         
    \     2       2  /           2*|3 + ---- + -----|         
                                   \     2       2  /
−i(eix−e−ix)(−eix2+3−e−ix2)2(eix2+3+e−ix2)2−i(eix−e−ix)2(eix2+3+e−ix2)- \frac{i \left(e^{i x} - e^{- i x}\right) \left(- \frac{e^{i x}}{2} + 3 - \frac{e^{- i x}}{2}\right)}{2 \left(\frac{e^{i x}}{2} + 3 + \frac{e^{- i x}}{2}\right)^{2}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(\frac{e^{i x}}{2} + 3 + \frac{e^{- i x}}{2}\right)} 
-i*(-exp(-i*x) + exp(i*x))/(2*(3 + exp(i*x)/2 + exp(-i*x)/2)) - i*(-exp(-i*x) + exp(i*x))*(3 - exp(i*x)/2 - exp(-i*x)/2)/(2*(3 + exp(i*x)/2 + exp(-i*x)/2)^2)
Denominador común [src] 
       6*sin(x)       
----------------------
       2              
9 + cos (x) + 6*cos(x)
6sin⁡(x)cos⁡2(x)+6cos⁡(x)+9\frac{6 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} + 6 \cos{\left(x \right)} + 9} 
6*sin(x)/(9 + cos(x)^2 + 6*cos(x))
Unión de expresiones racionales [src] 
   6*sin(x)  
-------------
            2
(3 + cos(x))
6sin⁡(x)(cos⁡(x)+3)2\frac{6 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} 
6*sin(x)/(3 + cos(x))^2
Abrimos la expresión [src] 
  sin(x)            3*sin(x)              cos(x)*sin(x)     
---------- + ---------------------- - ----------------------
3 + cos(x)          2                        2              
             9 + cos (x) + 6*cos(x)   9 + cos (x) + 6*cos(x)
−sin⁡(x)cos⁡(x)cos⁡2(x)+6cos⁡(x)+9+3sin⁡(x)cos⁡2(x)+6cos⁡(x)+9+sin⁡(x)cos⁡(x)+3- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\cos^{2}{\left(x \right)} + 6 \cos{\left(x \right)} + 9} + \frac{3 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} + 6 \cos{\left(x \right)} + 9} + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 3} 
sin(x)/(3 + cos(x)) + 3*sin(x)/(9 + cos(x)^2 + 6*cos(x)) - cos(x)*sin(x)/(9 + cos(x)^2 + 6*cos(x))
Parte trigonométrica [src] 
                                         1          
                                   3 - ------       
           1                           sec(x)       
------------------------ + -------------------------
/      1   \    /    pi\               2            
|3 + ------|*sec|x - --|   /      1   \     /    pi\
\    sec(x)/    \    2 /   |3 + ------| *sec|x - --|
                           \    sec(x)/     \    2 /
3−1sec⁡(x)(3+1sec⁡(x))2sec⁡(x−π2)+1(3+1sec⁡(x))sec⁡(x−π2)\frac{3 - \frac{1}{\sec{\left(x \right)}}}{\left(3 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(3 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
                                        /            2/x\\          
                                        |    -1 + cot |-||          
                                        |             \2/|    /x\   
                                      2*|3 - ------------|*cot|-|   
                 /x\                    |           2/x\ |    \2/   
            2*cot|-|                    |    1 + cot |-| |          
                 \2/                    \            \2/ /          
-------------------------------- + ---------------------------------
              /            2/x\\                                   2
              |    -1 + cot |-||                 /            2/x\\ 
/       2/x\\ |             \2/|                 |    -1 + cot |-|| 
|1 + cot |-||*|3 + ------------|   /       2/x\\ |             \2/| 
\        \2// |           2/x\ |   |1 + cot |-||*|3 + ------------| 
              |    1 + cot |-| |   \        \2// |           2/x\ | 
              \            \2/ /                 |    1 + cot |-| | 
                                                 \            \2/ /
2(−cot⁡2(x2)−1cot⁡2(x2)+1+3)cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+3)2(cot⁡2(x2)+1)+2cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+3)(cot⁡2(x2)+1)\frac{2 \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right) \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
     6*sin(x)     
------------------
                 2
/       /    pi\\ 
|3 + sin|x + --|| 
\       \    2 //
6sin⁡(x)(sin⁡(x+π2)+3)2\frac{6 \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 3\right)^{2}} 
     /    pi\
6*cos|x - --|
     \    2 /
-------------
            2
(3 + cos(x))
6cos⁡(x−π2)(cos⁡(x)+3)2\frac{6 \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} 
   6*sin(x)  
-------------
            2
(3 + cos(x))
6sin⁡(x)(cos⁡(x)+3)2\frac{6 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} 
            6            
-------------------------
            2            
/      1   \     /    pi\
|3 + ------| *sec|x - --|
\    sec(x)/     \    2 /
6(3+1sec⁡(x))2sec⁡(x−π2)\frac{6}{\left(3 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} 
            6            
-------------------------
                 2       
/         1     \        
|3 + -----------| *csc(x)
|       /pi    \|        
|    csc|-- - x||        
\       \2     //
6(3+1csc⁡(−x+π2))2csc⁡(x)\frac{6}{\left(3 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} 
                 /x\            
           12*tan|-|            
                 \2/            
--------------------------------
                               2
              /           2/x\\ 
              |    1 - tan |-|| 
/       2/x\\ |            \2/| 
|1 + tan |-||*|3 + -----------| 
\        \2// |           2/x\| 
              |    1 + tan |-|| 
              \            \2//
12tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+3)2(tan⁡2(x2)+1)\frac{12 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                                         1          
                                3 - -----------     
                                       /pi    \     
                                    csc|-- - x|     
           1                           \2     /     
------------------------ + -------------------------
/         1     \                           2       
|3 + -----------|*csc(x)   /         1     \        
|       /pi    \|          |3 + -----------| *csc(x)
|    csc|-- - x||          |       /pi    \|        
\       \2     //          |    csc|-- - x||        
                           \       \2     //
3−1csc⁡(−x+π2)(3+1csc⁡(−x+π2))2csc⁡(x)+1(3+1csc⁡(−x+π2))csc⁡(x)\frac{3 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(3 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(3 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} 
                                 1        
                           3 - ------     
         1                     sec(x)     
------------------- + --------------------
/      1   \                      2       
|3 + ------|*csc(x)   /      1   \        
\    sec(x)/          |3 + ------| *csc(x)
                      \    sec(x)/
3−1sec⁡(x)(3+1sec⁡(x))2csc⁡(x)+1(3+1sec⁡(x))csc⁡(x)\frac{3 - \frac{1}{\sec{\left(x \right)}}}{\left(3 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(3 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}} 
                                       /           2/x\\          
                                       |    1 - tan |-||          
                                       |            \2/|    /x\   
                                     2*|3 - -----------|*tan|-|   
                 /x\                   |           2/x\|    \2/   
            2*tan|-|                   |    1 + tan |-||          
                 \2/                   \            \2//          
------------------------------- + --------------------------------
              /           2/x\\                                  2
              |    1 - tan |-||                 /           2/x\\ 
/       2/x\\ |            \2/|                 |    1 - tan |-|| 
|1 + tan |-||*|3 + -----------|   /       2/x\\ |            \2/| 
\        \2// |           2/x\|   |1 + tan |-||*|3 + -----------| 
              |    1 + tan |-||   \        \2// |           2/x\| 
              \            \2//                 |    1 + tan |-|| 
                                                \            \2//
2(−1−tan⁡2(x2)tan⁡2(x2)+1+3)tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+3)2(tan⁡2(x2)+1)+2tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+3)(tan⁡2(x2)+1)\frac{2 \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right) \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                  /x\            
            12*cot|-|            
                  \2/            
---------------------------------
                                2
              /            2/x\\ 
              |    -1 + cot |-|| 
/       2/x\\ |             \2/| 
|1 + cot |-||*|3 + ------------| 
\        \2// |           2/x\ | 
              |    1 + cot |-| | 
              \            \2/ /
12cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+3)2(cot⁡2(x2)+1)\frac{12 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
   /    pi\                   /    pi\
cos|x - --|   (3 - cos(x))*cos|x - --|
   \    2 /                   \    2 /
----------- + ------------------------
 3 + cos(x)                    2      
                   (3 + cos(x))
(3−cos⁡(x))cos⁡(x−π2)(cos⁡(x)+3)2+cos⁡(x−π2)cos⁡(x)+3\frac{\left(3 - \cos{\left(x \right)}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 3\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 3} 
                  /       /    pi\\       
                  |3 - sin|x + --||*sin(x)
     sin(x)       \       \    2 //       
--------------- + ------------------------
       /    pi\                       2   
3 + sin|x + --|      /       /    pi\\    
       \    2 /      |3 + sin|x + --||    
                     \       \    2 //
(3−sin⁡(x+π2))sin⁡(x)(sin⁡(x+π2)+3)2+sin⁡(x)sin⁡(x+π2)+3\frac{\left(3 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 3\right)^{2}} + \frac{\sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 3} 
sin(x)/(3 + sin(x + pi/2)) + (3 - sin(x + pi/2))*sin(x)/(3 + sin(x + pi/2))^2
Denominador racional [src] 
            2                                          
(3 + cos(x)) *sin(x) + (3 - cos(x))*(3 + cos(x))*sin(x)
-------------------------------------------------------
                                 3                     
                     (3 + cos(x))
(3−cos⁡(x))(cos⁡(x)+3)sin⁡(x)+(cos⁡(x)+3)2sin⁡(x)(cos⁡(x)+3)3\frac{\left(3 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 3\right) \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 3\right)^{2} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 3\right)^{3}} 
((3 + cos(x))^2*sin(x) + (3 - cos(x))*(3 + cos(x))*sin(x))/(3 + cos(x))^3
    © Sr Examen – Калькулятор