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Resolución de integrales/ (3sin(x)+1)/(2sin(2x)+15cos(2x))

Integral de (3sin(x)+1)/(2sin(2x)+15cos(2x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 2*pi                           
   /                            
  |                             
  |        3*sin(x) + 1         
  |  ------------------------ dx
  |  2*sin(2*x) + 15*cos(2*x)   
  |                             
 /                              
 0
∫02π3sin⁡(x)+12sin⁡(2x)+15cos⁡(2x) dx\int\limits_{0}^{2 \pi} \frac{3 \sin{\left(x \right)} + 1}{2 \sin{\left(2 x \right)} + 15 \cos{\left(2 x \right)}}\, dx 
Integral((3*sin(x) + 1)/(2*sin(2*x) + 15*cos(2*x)), (x, 0, 2*pi))
Respuesta (Indefinida) [src] 
                                                                                   /         _____         \              /         _____         \
  /                                      /                                _____    |  2    \/ 229          |     _____    |  2    \/ 229          |
 |                                      |                               \/ 229 *log|- -- - ------- + tan(x)|   \/ 229 *log|- -- + ------- + tan(x)|
 |       3*sin(x) + 1                   |          sin(x)                          \  15      15           /              \  15      15           /
 | ------------------------ dx = C + 3* | ------------------------ dx - ------------------------------------ + ------------------------------------
 | 2*sin(2*x) + 15*cos(2*x)             | 2*sin(2*x) + 15*cos(2*x)                      458                                    458                 
 |                                      |                                                                                                          
/                                      /
∫3sin⁡(x)+12sin⁡(2x)+15cos⁡(2x) dx=C+229log⁡(tan⁡(x)−215+22915)458−229log⁡(tan⁡(x)−22915−215)458+3∫sin⁡(x)2sin⁡(2x)+15cos⁡(2x) dx\int \frac{3 \sin{\left(x \right)} + 1}{2 \sin{\left(2 x \right)} + 15 \cos{\left(2 x \right)}}\, dx = C + \frac{\sqrt{229} \log{\left(\tan{\left(x \right)} - \frac{2}{15} + \frac{\sqrt{229}}{15} \right)}}{458} - \frac{\sqrt{229} \log{\left(\tan{\left(x \right)} - \frac{\sqrt{229}}{15} - \frac{2}{15} \right)}}{458} + 3 \int \frac{\sin{\left(x \right)}}{2 \sin{\left(2 x \right)} + 15 \cos{\left(2 x \right)}}\, dx
Simplificar
Respuesta [src] 
 2*pi                           
   /                            
  |                             
  |        1 + 3*sin(x)         
  |  ------------------------ dx
  |  2*sin(2*x) + 15*cos(2*x)   
  |                             
 /                              
 0
∫02π3sin⁡(x)+12sin⁡(2x)+15cos⁡(2x) dx\int\limits_{0}^{2 \pi} \frac{3 \sin{\left(x \right)} + 1}{2 \sin{\left(2 x \right)} + 15 \cos{\left(2 x \right)}}\, dx 
=
= 
 2*pi                           
   /                            
  |                             
  |        1 + 3*sin(x)         
  |  ------------------------ dx
  |  2*sin(2*x) + 15*cos(2*x)   
  |                             
 /                              
 0
∫02π3sin⁡(x)+12sin⁡(2x)+15cos⁡(2x) dx\int\limits_{0}^{2 \pi} \frac{3 \sin{\left(x \right)} + 1}{2 \sin{\left(2 x \right)} + 15 \cos{\left(2 x \right)}}\, dx 
Integral((1 + 3*sin(x))/(2*sin(2*x) + 15*cos(2*x)), (x, 0, 2*pi))
Respuesta numérica [src] 
0.697142898472578
0.697142898472578

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

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