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Resolución de integrales/ cos(x)/ sin(x)/ sin(2x)/ sin(2x)*sin(x)/(1+cos(x))

Integral de sin(2x)*sin(x)/(1+cos(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |  sin(2*x)*sin(x)   
 |  --------------- dx
 |     1 + cos(x)     
 |                    
/                     
0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(x \right)} + 1}\, dx$$ 
Integral((sin(2*x)*sin(x))/(1 + cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                   /x\                         3/x\                        2/x\                        4/x\       
 |                                                               4*tan|-|                   12*tan |-|                 4*x*tan |-|                 2*x*tan |-|       
 | sin(2*x)*sin(x)                     2*x                            \2/                          \2/                         \2/                         \2/       
 | --------------- dx = C - ------------------------- + ------------------------- + ------------------------- - ------------------------- - -------------------------
 |    1 + cos(x)                     4/x\        2/x\            4/x\        2/x\            4/x\        2/x\            4/x\        2/x\            4/x\        2/x\
 |                          2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|
/                                     \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/
$$\int \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(x \right)} + 1}\, dx = C - \frac{2 x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{4 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 x}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                2                               4                                                                3             
                2                          4*tan (1/2)                     2*tan (1/2)                      4*tan(1/2)                     12*tan (1/2)        
- ----------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + -----------------------------
           4             2                 4             2                 4             2                 4             2                 4             2     
  2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)
$$- \frac{2}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} - \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} - \frac{2 \tan^{4}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2}$$ 
=
= 
                                                2                               4                                                                3             
                2                          4*tan (1/2)                     2*tan (1/2)                      4*tan(1/2)                     12*tan (1/2)        
- ----------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + -----------------------------
           4             2                 4             2                 4             2                 4             2                 4             2     
  2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)
$$- \frac{2}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} - \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} - \frac{2 \tan^{4}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2}$$ 
-2/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) - 4*tan(1/2)^2/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) - 2*tan(1/2)^4/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) + 4*tan(1/2)/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) + 12*tan(1/2)^3/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2)
Respuesta numérica [src] 
0.228293256202952
0.228293256202952

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

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