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Resolución de integrales/ cos(x)/ sin(x)/ 1/(3*sin(x)+5*cos(x))

Integral de 1/(3*sin(x)+5*cos(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                       
 --                       
 2                        
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |  3*sin(x) + 5*cos(x)   
 |                        
/                         
0
$$\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3 \sin{\left(x \right)} + 5 \cos{\left(x \right)}}\, dx$$ 
Integral(1/(3*sin(x) + 5*cos(x)), (x, 0, pi/2))
Respuesta (Indefinida) [src] 
                                          /        ____         \             /        ____         \
  /                               ____    |  3   \/ 34       /x\|     ____    |  3   \/ 34       /x\|
 |                              \/ 34 *log|- - - ------ + tan|-||   \/ 34 *log|- - + ------ + tan|-||
 |          1                             \  5     5         \2//             \  5     5         \2//
 | ------------------- dx = C - --------------------------------- + ---------------------------------
 | 3*sin(x) + 5*cos(x)                          34                                  34               
 |                                                                                                   
/
$$\int \frac{1}{3 \sin{\left(x \right)} + 5 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{34} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{5} + \frac{\sqrt{34}}{5} \right)}}{34} - \frac{\sqrt{34} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{34}}{5} - \frac{3}{5} \right)}}{34}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
         /          /        ____\\             /        ____\          /          /      ____\\             /      ____\
    ____ |          |  2   \/ 34 ||     ____    |  3   \/ 34 |     ____ |          |3   \/ 34 ||     ____    |2   \/ 34 |
  \/ 34 *|pi*I + log|- - + ------||   \/ 34 *log|- - + ------|   \/ 34 *|pi*I + log|- + ------||   \/ 34 *log|- + ------|
         \          \  5     5   //             \  5     5   /          \          \5     5   //             \5     5   /
- --------------------------------- - ------------------------ + ------------------------------- + ----------------------
                  34                             34                             34                           34
$$\frac{\sqrt{34} \log{\left(\frac{2}{5} + \frac{\sqrt{34}}{5} \right)}}{34} - \frac{\sqrt{34} \log{\left(- \frac{3}{5} + \frac{\sqrt{34}}{5} \right)}}{34} - \frac{\sqrt{34} \left(\log{\left(- \frac{2}{5} + \frac{\sqrt{34}}{5} \right)} + i \pi\right)}{34} + \frac{\sqrt{34} \left(\log{\left(\frac{3}{5} + \frac{\sqrt{34}}{5} \right)} + i \pi\right)}{34}$$ 
=
= 
         /          /        ____\\             /        ____\          /          /      ____\\             /      ____\
    ____ |          |  2   \/ 34 ||     ____    |  3   \/ 34 |     ____ |          |3   \/ 34 ||     ____    |2   \/ 34 |
  \/ 34 *|pi*I + log|- - + ------||   \/ 34 *log|- - + ------|   \/ 34 *|pi*I + log|- + ------||   \/ 34 *log|- + ------|
         \          \  5     5   //             \  5     5   /          \          \5     5   //             \5     5   /
- --------------------------------- - ------------------------ + ------------------------------- + ----------------------
                  34                             34                             34                           34
$$\frac{\sqrt{34} \log{\left(\frac{2}{5} + \frac{\sqrt{34}}{5} \right)}}{34} - \frac{\sqrt{34} \log{\left(- \frac{3}{5} + \frac{\sqrt{34}}{5} \right)}}{34} - \frac{\sqrt{34} \left(\log{\left(- \frac{2}{5} + \frac{\sqrt{34}}{5} \right)} + i \pi\right)}{34} + \frac{\sqrt{34} \left(\log{\left(\frac{3}{5} + \frac{\sqrt{34}}{5} \right)} + i \pi\right)}{34}$$ 
-sqrt(34)*(pi*i + log(-2/5 + sqrt(34)/5))/34 - sqrt(34)*log(-3/5 + sqrt(34)/5)/34 + sqrt(34)*(pi*i + log(3/5 + sqrt(34)/5))/34 + sqrt(34)*log(2/5 + sqrt(34)/5)/34
Respuesta numérica [src] 
0.317721805098958
0.317721805098958

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

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