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Simplificación de expresiones/ Descomposición en fracciones simples/ sin/(a*x-1)-1/(a*x+1)

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

Expresión a simplificar:

Solución

Ha introducido [src] 
 sin(x)      1   
------- - -------
a*x - 1   a*x + 1
$$- \frac{1}{a x + 1} + \frac{\sin{\left(x \right)}}{a x - 1}$$ 
sin(x)/(a*x - 1) - 1/(a*x + 1)
Simplificación general [src] 
1 + (1 + a*x)*sin(x) - a*x
--------------------------
   (1 + a*x)*(-1 + a*x)
$$\frac{- a x + \left(a x + 1\right) \sin{\left(x \right)} + 1}{\left(a x - 1\right) \left(a x + 1\right)}$$ 
(1 + (1 + a*x)*sin(x) - a*x)/((1 + a*x)*(-1 + a*x))
Unión de expresiones racionales [src] 
1 + (1 + a*x)*sin(x) - a*x
--------------------------
   (1 + a*x)*(-1 + a*x)
$$\frac{- a x + \left(a x + 1\right) \sin{\left(x \right)} + 1}{\left(a x - 1\right) \left(a x + 1\right)}$$ 
(1 + (1 + a*x)*sin(x) - a*x)/((1 + a*x)*(-1 + a*x))
Combinatoria [src] 
1 - a*x + a*x*sin(x) + sin(x)
-----------------------------
     (1 + a*x)*(-1 + a*x)
$$\frac{a x \sin{\left(x \right)} - a x + \sin{\left(x \right)} + 1}{\left(a x - 1\right) \left(a x + 1\right)}$$ 
(1 - a*x + a*x*sin(x) + sin(x))/((1 + a*x)*(-1 + a*x))
Potencias [src] 
              /   -I*x    I*x\
     1      I*\- e     + e   /
- ------- - ------------------
  1 + a*x      2*(-1 + a*x)
$$- \frac{1}{a x + 1} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(a x - 1\right)}$$ 
-1/(1 + a*x) - i*(-exp(-i*x) + exp(i*x))/(2*(-1 + a*x))
Denominador racional [src] 
1 + (1 + a*x)*sin(x) - a*x
--------------------------
   (1 + a*x)*(-1 + a*x)
$$\frac{- a x + \left(a x + 1\right) \sin{\left(x \right)} + 1}{\left(a x - 1\right) \left(a x + 1\right)}$$ 
(1 + (1 + a*x)*sin(x) - a*x)/((1 + a*x)*(-1 + a*x))
Denominador común [src] 
1 - a*x + a*x*sin(x) + sin(x)
-----------------------------
                2  2         
          -1 + a *x
$$\frac{a x \sin{\left(x \right)} - a x + \sin{\left(x \right)} + 1}{a^{2} x^{2} - 1}$$ 
(1 - a*x + a*x*sin(x) + sin(x))/(-1 + a^2*x^2)
Respuesta numérica [src] 
-1/(1.0 + a*x) + sin(x)/(-1.0 + a*x)
-1/(1.0 + a*x) + sin(x)/(-1.0 + a*x)
Parte trigonométrica [src] 
                         /x\        
                    2*cot|-|        
     1                   \2/        
- ------- + ------------------------
  1 + a*x   /       2/x\\           
            |1 + cot |-||*(-1 + a*x)
            \        \2//
$$- \frac{1}{a x + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(a x - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
               /    pi\
            cos|x - --|
     1         \    2 /
- ------- + -----------
  1 + a*x     -1 + a*x
$$- \frac{1}{a x + 1} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{a x - 1}$$ 
     1              1        
- ------- + -----------------
  1 + a*x   (-1 + a*x)*csc(x)
$$- \frac{1}{a x + 1} + \frac{1}{\left(a x - 1\right) \csc{\left(x \right)}}$$ 
     1                1           
- ------- + ----------------------
  1 + a*x                 /    pi\
            (-1 + a*x)*sec|x - --|
                          \    2 /
$$- \frac{1}{a x + 1} + \frac{1}{\left(a x - 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$ 
                         /x\        
                    2*tan|-|        
     1                   \2/        
- ------- + ------------------------
  1 + a*x   /       2/x\\           
            |1 + tan |-||*(-1 + a*x)
            \        \2//
$$- \frac{1}{a x + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(a x - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
-1/(1 + a*x) + 2*tan(x/2)/((1 + tan(x/2)^2)*(-1 + a*x))
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