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Resolución de integrales/ xsinx/ cosx+x/ x^2/(cosx+xsinx)^2

Integral de x^2/(cosx+xsinx)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |            2            
 |           x             
 |  -------------------- dx
 |                     2   
 |  (cos(x) + x*sin(x))    
 |                         
/                          
0
$$\int\limits_{0}^{1} \frac{x^{2}}{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx$$ 
Integral(x^2/(cos(x) + x*sin(x))^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                            
 |                                                                       /x\                       2/x\        
 |           2                                                      2*tan|-|                  x*tan |-|        
 |          x                               x                            \2/                        \2/        
 | -------------------- dx = C - ------------------------ + ------------------------ + ------------------------
 |                    2                 2/x\          /x\          2/x\          /x\          2/x\          /x\
 | (cos(x) + x*sin(x))           1 - tan |-| + 2*x*tan|-|   1 - tan |-| + 2*x*tan|-|   1 - tan |-| + 2*x*tan|-|
 |                                       \2/          \2/           \2/          \2/           \2/          \2/
/
$$\int \frac{x^{2}}{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx = C + \frac{x \tan^{2}{\left(\frac{x}{2} \right)}}{2 x \tan{\left(\frac{x}{2} \right)} - \tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{x}{2 x \tan{\left(\frac{x}{2} \right)} - \tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 x \tan{\left(\frac{x}{2} \right)} - \tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                          2                                           
              1                        tan (1/2)                    2*tan(1/2)        
- -------------------------- + -------------------------- + --------------------------
         2                            2                            2                  
  1 - tan (1/2) + 2*tan(1/2)   1 - tan (1/2) + 2*tan(1/2)   1 - tan (1/2) + 2*tan(1/2)
$$- \frac{1}{- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)}} + \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \tan{\left(\frac{1}{2} \right)}}{- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)}}$$ 
=
= 
                                          2                                           
              1                        tan (1/2)                    2*tan(1/2)        
- -------------------------- + -------------------------- + --------------------------
         2                            2                            2                  
  1 - tan (1/2) + 2*tan(1/2)   1 - tan (1/2) + 2*tan(1/2)   1 - tan (1/2) + 2*tan(1/2)
$$- \frac{1}{- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)}} + \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \tan{\left(\frac{1}{2} \right)}}{- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)}}$$ 
-1/(1 - tan(1/2)^2 + 2*tan(1/2)) + tan(1/2)^2/(1 - tan(1/2)^2 + 2*tan(1/2)) + 2*tan(1/2)/(1 - tan(1/2)^2 + 2*tan(1/2))
Respuesta numérica [src] 
0.217958098460862
0.217958098460862

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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