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Resolución de integrales/ 8sinx/ cosx+3/ 1/(8sinx-7cosx+3)

Integral de 1/(8sinx-7cosx+3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |             1              
 |  ----------------------- dx
 |  8*sin(x) - 7*cos(x) + 3   
 |                            
/                             
0
∫011(8sin⁡(x)−7cos⁡(x))+3 dx\int\limits_{0}^{1} \frac{1}{\left(8 \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) + 3}\, dx 
Integral(1/(8*sin(x) - 7*cos(x) + 3), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    1(8sin⁡(x)−7cos⁡(x))+3=−1−8sin⁡(x)+7cos⁡(x)−3\frac{1}{\left(8 \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) + 3} = - \frac{1}{- 8 \sin{\left(x \right)} + 7 \cos{\left(x \right)} - 3}

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

    ∫(−1−8sin⁡(x)+7cos⁡(x)−3) dx=−∫1−8sin⁡(x)+7cos⁡(x)−3 dx\int \left(- \frac{1}{- 8 \sin{\left(x \right)} + 7 \cos{\left(x \right)} - 3}\right)\, dx = - \int \frac{1}{- 8 \sin{\left(x \right)} + 7 \cos{\left(x \right)} - 3}\, dx

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

      Pero la integral

      26log⁡(tan⁡(x2)+45+265)52−26log⁡(tan⁡(x2)−265+45)52\frac{\sqrt{26} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} - \frac{\sqrt{26} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{26}}{5} + \frac{4}{5} \right)}}{52}

    Por lo tanto, el resultado es: −26log⁡(tan⁡(x2)+45+265)52+26log⁡(tan⁡(x2)−265+45)52- \frac{\sqrt{26} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} + \frac{\sqrt{26} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{26}}{5} + \frac{4}{5} \right)}}{52}

  3. Ahora simplificar:

    26(−log⁡(tan⁡(x2)+45+265)+log⁡(tan⁡(x2)−265+45))52\frac{\sqrt{26} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{26}}{5} + \frac{4}{5} \right)}\right)}{52}

  4. Añadimos la constante de integración:

    26(−log⁡(tan⁡(x2)+45+265)+log⁡(tan⁡(x2)−265+45))52+constant\frac{\sqrt{26} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{26}}{5} + \frac{4}{5} \right)}\right)}{52}+ \mathrm{constant}

Respuesta:

26(−log⁡(tan⁡(x2)+45+265)+log⁡(tan⁡(x2)−265+45))52+constant\frac{\sqrt{26} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{26}}{5} + \frac{4}{5} \right)}\right)}{52}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                              /      ____         \             /      ____         \
  /                                   ____    |4   \/ 26       /x\|     ____    |4   \/ 26       /x\|
 |                                  \/ 26 *log|- + ------ + tan|-||   \/ 26 *log|- - ------ + tan|-||
 |            1                               \5     5         \2//             \5     5         \2//
 | ----------------------- dx = C - ------------------------------- + -------------------------------
 | 8*sin(x) - 7*cos(x) + 3                         52                                52              
 |                                                                                                   
/
∫1(8sin⁡(x)−7cos⁡(x))+3 dx=C−26log⁡(tan⁡(x2)+45+265)52+26log⁡(tan⁡(x2)−265+45)52\int \frac{1}{\left(8 \sin{\left(x \right)} - 7 \cos{\left(x \right)}\right) + 3}\, dx = C - \frac{\sqrt{26} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} + \frac{\sqrt{26} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{26}}{5} + \frac{4}{5} \right)}}{52}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1000010000
Trazar el gráfico f(x)
Respuesta [src] 
         /          /        ____\\             /      ____           \             /      ____\             /      ____           \
    ____ |          |  4   \/ 26 ||     ____    |4   \/ 26            |     ____    |4   \/ 26 |     ____    |4   \/ 26            |
  \/ 26 *|pi*I + log|- - + ------||   \/ 26 *log|- + ------ + tan(1/2)|   \/ 26 *log|- + ------|   \/ 26 *log|- - ------ + tan(1/2)|
         \          \  5     5   //             \5     5              /             \5     5   /             \5     5              /
- --------------------------------- - --------------------------------- + ---------------------- + ---------------------------------
                  52                                  52                            52                             52
26log⁡(−265+tan⁡(12)+45)52−26log⁡(tan⁡(12)+45+265)52+26log⁡(45+265)52−26(log⁡(−45+265)+iπ)52\frac{\sqrt{26} \log{\left(- \frac{\sqrt{26}}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{4}{5} \right)}}{52} - \frac{\sqrt{26} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} + \frac{\sqrt{26} \log{\left(\frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} - \frac{\sqrt{26} \left(\log{\left(- \frac{4}{5} + \frac{\sqrt{26}}{5} \right)} + i \pi\right)}{52} 
=
= 
         /          /        ____\\             /      ____           \             /      ____\             /      ____           \
    ____ |          |  4   \/ 26 ||     ____    |4   \/ 26            |     ____    |4   \/ 26 |     ____    |4   \/ 26            |
  \/ 26 *|pi*I + log|- - + ------||   \/ 26 *log|- + ------ + tan(1/2)|   \/ 26 *log|- + ------|   \/ 26 *log|- - ------ + tan(1/2)|
         \          \  5     5   //             \5     5              /             \5     5   /             \5     5              /
- --------------------------------- - --------------------------------- + ---------------------- + ---------------------------------
                  52                                  52                            52                             52
26log⁡(−265+tan⁡(12)+45)52−26log⁡(tan⁡(12)+45+265)52+26log⁡(45+265)52−26(log⁡(−45+265)+iπ)52\frac{\sqrt{26} \log{\left(- \frac{\sqrt{26}}{5} + \tan{\left(\frac{1}{2} \right)} + \frac{4}{5} \right)}}{52} - \frac{\sqrt{26} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} + \frac{\sqrt{26} \log{\left(\frac{4}{5} + \frac{\sqrt{26}}{5} \right)}}{52} - \frac{\sqrt{26} \left(\log{\left(- \frac{4}{5} + \frac{\sqrt{26}}{5} \right)} + i \pi\right)}{52} 
-sqrt(26)*(pi*i + log(-4/5 + sqrt(26)/5))/52 - sqrt(26)*log(4/5 + sqrt(26)/5 + tan(1/2))/52 + sqrt(26)*log(4/5 + sqrt(26)/5)/52 + sqrt(26)*log(4/5 - sqrt(26)/5 + tan(1/2))/52
Respuesta numérica [src] 
0.00475582122394907
0.00475582122394907

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

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