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Resolución de integrales/ 1/((sin(x)^2+8*sin(x)*cos(x)+12*(cos(x))^2))

Integral de 1/((sin(x)^2+8*sin(x)*cos(x)+12*(cos(x))^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                                          
  /                                          
 |                                           
 |                    1                      
 |  -------------------------------------- dx
 |     2                              2      
 |  sin (x) + 8*sin(x)*cos(x) + 12*cos (x)   
 |                                           
/                                            
0
$$\int\limits_{0}^{1} \frac{1}{\left(\sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) + 12 \cos^{2}{\left(x \right)}}\, dx$$ 
Integral(1/(sin(x)^2 + (8*sin(x))*cos(x) + 12*cos(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                   /        /x\        2/x\\      /        2/x\      /x\\
 |                                                 log|-3 - tan|-| + 3*tan |-||   log|-1 + tan |-| - tan|-||
 |                   1                                \        \2/         \2//      \         \2/      \2//
 | -------------------------------------- dx = C - ---------------------------- + --------------------------
 |    2                              2                          4                             4             
 | sin (x) + 8*sin(x)*cos(x) + 12*cos (x)                                                                   
 |                                                                                                          
/
$$\int \frac{1}{\left(\sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) + 12 \cos^{2}{\left(x \right)}}\, dx = C + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\log{\left(3 \tan^{2}{\left(\frac{x}{2} \right)} - \tan{\left(\frac{x}{2} \right)} - 3 \right)}}{4}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
     /         2                \               /       2                \
  log\3 - 3*tan (1/2) + tan(1/2)/   log(3)   log\1 - tan (1/2) + tan(1/2)/
- ------------------------------- + ------ + -----------------------------
                 4                    4                    4
$$- \frac{\log{\left(- 3 \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)}}{4} + \frac{\log{\left(3 \right)}}{4}$$ 
=
= 
     /         2                \               /       2                \
  log\3 - 3*tan (1/2) + tan(1/2)/   log(3)   log\1 - tan (1/2) + tan(1/2)/
- ------------------------------- + ------ + -----------------------------
                 4                    4                    4
$$- \frac{\log{\left(- 3 \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{4} + \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)}}{4} + \frac{\log{\left(3 \right)}}{4}$$ 
-log(3 - 3*tan(1/2)^2 + tan(1/2))/4 + log(3)/4 + log(1 - tan(1/2)^2 + tan(1/2))/4
Respuesta numérica [src] 
0.0862790408079794
0.0862790408079794

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

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