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Resolución de integrales/ sin(x)/ sin(3x)/ sin(x)/sin(3x)

Integral de sin(x)/sin(3x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |   sin(x)    
 |  -------- dx
 |  sin(3*x)   
 |             
/              
0
∫01sin⁡(x)sin⁡(3x) dx\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{\sin{\left(3 x \right)}}\, dx 
Integral(sin(x)/sin(3*x), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    sin⁡(x)sin⁡(3x)=sin⁡(x)−4sin⁡3(x)+3sin⁡(x)\frac{\sin{\left(x \right)}}{\sin{\left(3 x \right)}} = \frac{\sin{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}}

  2. Vuelva a escribir el integrando:

    sin⁡(x)−4sin⁡3(x)+3sin⁡(x)=−14sin⁡2(x)−3\frac{\sin{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}} = - \frac{1}{4 \sin^{2}{\left(x \right)} - 3}

  3. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫(−14sin⁡2(x)−3) dx=−∫14sin⁡2(x)−3 dx\int \left(- \frac{1}{4 \sin^{2}{\left(x \right)} - 3}\right)\, dx = - \int \frac{1}{4 \sin^{2}{\left(x \right)} - 3}\, dx

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

      −3log⁡(tan⁡(x2)−3)6+3log⁡(tan⁡(x2)−33)6−3log⁡(tan⁡(x2)+33)6+3log⁡(tan⁡(x2)+3)6- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}}{6}

    Por lo tanto, el resultado es: 3log⁡(tan⁡(x2)−3)6−3log⁡(tan⁡(x2)−33)6+3log⁡(tan⁡(x2)+33)6−3log⁡(tan⁡(x2)+3)6\frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}}{6}

  4. Ahora simplificar:

    3(log⁡(tan⁡(x2)−3)−log⁡(tan⁡(x2)−33)+log⁡(tan⁡(x2)+33)−log⁡(tan⁡(x2)+3))6\frac{\sqrt{3} \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}\right)}{6}

  5. Añadimos la constante de integración:

    3(log⁡(tan⁡(x2)−3)−log⁡(tan⁡(x2)−33)+log⁡(tan⁡(x2)+33)−log⁡(tan⁡(x2)+3))6+constant\frac{\sqrt{3} \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}\right)}{6}+ \mathrm{constant}

Respuesta:

3(log⁡(tan⁡(x2)−3)−log⁡(tan⁡(x2)−33)+log⁡(tan⁡(x2)+33)−log⁡(tan⁡(x2)+3))6+constant\frac{\sqrt{3} \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}\right)}{6}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                                          /    ___         \                                          /  ___         \
  /                    ___    /  ___      /x\\     ___    |  \/ 3       /x\|     ___    /    ___      /x\\     ___    |\/ 3       /x\|
 |                   \/ 3 *log|\/ 3  + tan|-||   \/ 3 *log|- ----- + tan|-||   \/ 3 *log|- \/ 3  + tan|-||   \/ 3 *log|----- + tan|-||
 |  sin(x)                    \           \2//            \    3        \2//            \             \2//            \  3        \2//
 | -------- dx = C - ------------------------- - --------------------------- + --------------------------- + -------------------------
 | sin(3*x)                      6                            6                             6                            6            
 |                                                                                                                                    
/
∫sin⁡(x)sin⁡(3x) dx=C+3log⁡(tan⁡(x2)−3)6−3log⁡(tan⁡(x2)−33)6+3log⁡(tan⁡(x2)+33)6−3log⁡(tan⁡(x2)+3)6\int \frac{\sin{\left(x \right)}}{\sin{\left(3 x \right)}}\, dx = C + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}}{6}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90010
Trazar el gráfico f(x)
Respuesta [src] 
                                    /          /              ___\\            /  ___\                                       /          /  ___\\                                                                      /  ___           \
                                ___ |          |            \/ 3 ||     ___    |\/ 3 |                                   ___ |          |\/ 3 ||                                                               ___    |\/ 3            |
    ___ /          /  ___\\   \/ 3 *|pi*I + log|-tan(1/2) + -----||   \/ 3 *log|-----|     ___    /  ___           \   \/ 3 *|pi*I + log|-----||     ___ /          /  ___           \\     ___    /  ___\   \/ 3 *log|----- + tan(1/2)|
  \/ 3 *\pi*I + log\\/ 3 //         \          \              3  //            \  3  /   \/ 3 *log\\/ 3  + tan(1/2)/         \          \  3  //   \/ 3 *\pi*I + log\\/ 3  - tan(1/2)//   \/ 3 *log\\/ 3 /            \  3             /
- ------------------------- - ------------------------------------- - ---------------- - --------------------------- + ------------------------- + ------------------------------------ + ---------------- + ---------------------------
              6                                 6                            6                        6                            6                                6                            6                        6
−3log⁡(tan⁡(12)+3)6+3log⁡(tan⁡(12)+33)6−3log⁡(33)6+3log⁡(3)6−3(log⁡(3)+iπ)6−3(log⁡(−tan⁡(12)+33)+iπ)6+3(log⁡(33)+iπ)6+3(log⁡(−tan⁡(12)+3)+iπ)6- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{6} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{6} - \frac{\sqrt{3} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{3}}{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(\frac{\sqrt{3}}{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{3} \right)} + i \pi\right)}{6} 
=
= 
                                    /          /              ___\\            /  ___\                                       /          /  ___\\                                                                      /  ___           \
                                ___ |          |            \/ 3 ||     ___    |\/ 3 |                                   ___ |          |\/ 3 ||                                                               ___    |\/ 3            |
    ___ /          /  ___\\   \/ 3 *|pi*I + log|-tan(1/2) + -----||   \/ 3 *log|-----|     ___    /  ___           \   \/ 3 *|pi*I + log|-----||     ___ /          /  ___           \\     ___    /  ___\   \/ 3 *log|----- + tan(1/2)|
  \/ 3 *\pi*I + log\\/ 3 //         \          \              3  //            \  3  /   \/ 3 *log\\/ 3  + tan(1/2)/         \          \  3  //   \/ 3 *\pi*I + log\\/ 3  - tan(1/2)//   \/ 3 *log\\/ 3 /            \  3             /
- ------------------------- - ------------------------------------- - ---------------- - --------------------------- + ------------------------- + ------------------------------------ + ---------------- + ---------------------------
              6                                 6                            6                        6                            6                                6                            6                        6
−3log⁡(tan⁡(12)+3)6+3log⁡(tan⁡(12)+33)6−3log⁡(33)6+3log⁡(3)6−3(log⁡(3)+iπ)6−3(log⁡(−tan⁡(12)+33)+iπ)6+3(log⁡(33)+iπ)6+3(log⁡(−tan⁡(12)+3)+iπ)6- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{6} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{6} - \frac{\sqrt{3} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{3}}{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(\frac{\sqrt{3}}{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{3} \right)} + i \pi\right)}{6} 
-sqrt(3)*(pi*i + log(sqrt(3)))/6 - sqrt(3)*(pi*i + log(-tan(1/2) + sqrt(3)/3))/6 - sqrt(3)*log(sqrt(3)/3)/6 - sqrt(3)*log(sqrt(3) + tan(1/2))/6 + sqrt(3)*(pi*i + log(sqrt(3)/3))/6 + sqrt(3)*(pi*i + log(sqrt(3) - tan(1/2)))/6 + sqrt(3)*log(sqrt(3))/6 + sqrt(3)*log(sqrt(3)/3 + tan(1/2))/6
Respuesta numérica [src] 
0.847473374471116
0.847473374471116

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

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