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Resolución de integrales/ cos(x)/ sin(x)/ dx/(sin(x)+2cos(x))

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

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