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Resolución de integrales/ 5cosx/ (1)/(1+5cosx)

Integral de (1)/(1+5cosx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |  1 + 5*cos(x)   
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{1}{5 \cos{\left(x \right)} + 1}\, dx$$ 
Integral(1/(1 + 5*cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                  /    ___         \            /  ___         \
  /                        ___    |  \/ 6       /x\|     ___    |\/ 6       /x\|
 |                       \/ 6 *log|- ----- + tan|-||   \/ 6 *log|----- + tan|-||
 |      1                         \    2        \2//            \  2        \2//
 | ------------ dx = C - --------------------------- + -------------------------
 | 1 + 5*cos(x)                       12                           12           
 |                                                                              
/
$$\int \frac{1}{5 \cos{\left(x \right)} + 1}\, dx = C - \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{6}}{2} \right)}}{12}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
        /          /  ___           \\            /  ___\         /          /  ___\\            /  ___           \
    ___ |          |\/ 6            ||     ___    |\/ 6 |     ___ |          |\/ 6 ||     ___    |\/ 6            |
  \/ 6 *|pi*I + log|----- - tan(1/2)||   \/ 6 *log|-----|   \/ 6 *|pi*I + log|-----||   \/ 6 *log|----- + tan(1/2)|
        \          \  2             //            \  2  /         \          \  2  //            \  2             /
- ------------------------------------ - ---------------- + ------------------------- + ---------------------------
                   12                           12                      12                           12
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$ 
=
= 
        /          /  ___           \\            /  ___\         /          /  ___\\            /  ___           \
    ___ |          |\/ 6            ||     ___    |\/ 6 |     ___ |          |\/ 6 ||     ___    |\/ 6            |
  \/ 6 *|pi*I + log|----- - tan(1/2)||   \/ 6 *log|-----|   \/ 6 *|pi*I + log|-----||   \/ 6 *log|----- + tan(1/2)|
        \          \  2             //            \  2  /         \          \  2  //            \  2             /
- ------------------------------------ - ---------------- + ------------------------- + ---------------------------
                   12                           12                      12                           12
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$ 
-sqrt(6)*(pi*i + log(sqrt(6)/2 - tan(1/2)))/12 - sqrt(6)*log(sqrt(6)/2)/12 + sqrt(6)*(pi*i + log(sqrt(6)/2))/12 + sqrt(6)*log(sqrt(6)/2 + tan(1/2))/12
Respuesta numérica [src] 
0.195862592929153
0.195862592929153

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

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