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Integral de 1/(2*cosx+3*sinx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                       
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |  2*cos(x) + 3*sin(x)   
 |                        
/                         
0                         
$$\int\limits_{0}^{1} \frac{1}{3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}\, dx$$
Integral(1/(2*cos(x) + 3*sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src]
                                          /        ____         \             /        ____         \
  /                               ____    |  3   \/ 13       /x\|     ____    |  3   \/ 13       /x\|
 |                              \/ 13 *log|- - - ------ + tan|-||   \/ 13 *log|- - + ------ + tan|-||
 |          1                             \  2     2         \2//             \  2     2         \2//
 | ------------------- dx = C - --------------------------------- + ---------------------------------
 | 2*cos(x) + 3*sin(x)                          13                                  13               
 |                                                                                                   
/                                                                                                    
$$\int \frac{1}{3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}}\, dx = C + \frac{\sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{3}{2} + \frac{\sqrt{13}}{2} \right)}}{13} - \frac{\sqrt{13} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{13}}{2} - \frac{3}{2} \right)}}{13}$$
Gráfica
Respuesta [src]
         /          /      ____           \\             /        ____\          /          /      ____\\             /        ____           \
    ____ |          |3   \/ 13            ||     ____    |  3   \/ 13 |     ____ |          |3   \/ 13 ||     ____    |  3   \/ 13            |
  \/ 13 *|pi*I + log|- + ------ - tan(1/2)||   \/ 13 *log|- - + ------|   \/ 13 *|pi*I + log|- + ------||   \/ 13 *log|- - + ------ + tan(1/2)|
         \          \2     2              //             \  2     2   /          \          \2     2   //             \  2     2              /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
                      13                                  13                             13                                  13                
$$\frac{\sqrt{13} \log{\left(- \frac{3}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{2} \right)}}{13} - \frac{\sqrt{13} \log{\left(- \frac{3}{2} + \frac{\sqrt{13}}{2} \right)}}{13} - \frac{\sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{3}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right)}{13} + \frac{\sqrt{13} \left(\log{\left(\frac{3}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right)}{13}$$
=
=
         /          /      ____           \\             /        ____\          /          /      ____\\             /        ____           \
    ____ |          |3   \/ 13            ||     ____    |  3   \/ 13 |     ____ |          |3   \/ 13 ||     ____    |  3   \/ 13            |
  \/ 13 *|pi*I + log|- + ------ - tan(1/2)||   \/ 13 *log|- - + ------|   \/ 13 *|pi*I + log|- + ------||   \/ 13 *log|- - + ------ + tan(1/2)|
         \          \2     2              //             \  2     2   /          \          \2     2   //             \  2     2              /
- ------------------------------------------ - ------------------------ + ------------------------------- + -----------------------------------
                      13                                  13                             13                                  13                
$$\frac{\sqrt{13} \log{\left(- \frac{3}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{13}}{2} \right)}}{13} - \frac{\sqrt{13} \log{\left(- \frac{3}{2} + \frac{\sqrt{13}}{2} \right)}}{13} - \frac{\sqrt{13} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{3}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right)}{13} + \frac{\sqrt{13} \left(\log{\left(\frac{3}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right)}{13}$$
-sqrt(13)*(pi*i + log(3/2 + sqrt(13)/2 - tan(1/2)))/13 - sqrt(13)*log(-3/2 + sqrt(13)/2)/13 + sqrt(13)*(pi*i + log(3/2 + sqrt(13)/2))/13 + sqrt(13)*log(-3/2 + sqrt(13)/2 + tan(1/2))/13
Respuesta numérica [src]
0.336140093570052
0.336140093570052

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