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Resolución de integrales/ 8sinx/ xcosx/ sin^2/ sinxcos/ 1/(19sin^2(x)-8sinxcosx-3)

Integral de 1/(19sin^2(x)-8sinxcosx-3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                                    
  /                                    
 |                                     
 |                 1                   
 |  -------------------------------- dx
 |        2                            
 |  19*sin (x) - 8*sin(x)*cos(x) - 3   
 |                                     
/                                      
0
∫011(19sin⁡2(x)−8sin⁡(x)cos⁡(x))−3 dx\int\limits_{0}^{1} \frac{1}{\left(19 \sin^{2}{\left(x \right)} - 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) - 3}\, dx 
Integral(1/(19*sin(x)^2 - 8*sin(x)*cos(x) - 3), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    1(19sin⁡2(x)−8sin⁡(x)cos⁡(x))−3=−1−19sin⁡2(x)+8sin⁡(x)cos⁡(x)+3\frac{1}{\left(19 \sin^{2}{\left(x \right)} - 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) - 3} = - \frac{1}{- 19 \sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \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:

    ∫(−1−19sin⁡2(x)+8sin⁡(x)cos⁡(x)+3) dx=−∫1−19sin⁡2(x)+8sin⁡(x)cos⁡(x)+3 dx\int \left(- \frac{1}{- 19 \sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)} + 3}\right)\, dx = - \int \frac{1}{- 19 \sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)} + 3}\, dx

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

      Pero la integral

      ∫1−19sin⁡2(x)+8sin⁡(x)cos⁡(x)+3 dx\int \frac{1}{- 19 \sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)} + 3}\, dx

    Por lo tanto, el resultado es: −∫1−19sin⁡2(x)+8sin⁡(x)cos⁡(x)+3 dx- \int \frac{1}{- 19 \sin^{2}{\left(x \right)} + 8 \sin{\left(x \right)} \cos{\left(x \right)} + 3}\, dx

  3. Ahora simplificar:

    −∫1−19sin⁡2(x)+4sin⁡(2x)+3 dx- \int \frac{1}{- 19 \sin^{2}{\left(x \right)} + 4 \sin{\left(2 x \right)} + 3}\, dx

  4. Añadimos la constante de integración:

    −∫1−19sin⁡2(x)+4sin⁡(2x)+3 dx+constant- \int \frac{1}{- 19 \sin^{2}{\left(x \right)} + 4 \sin{\left(2 x \right)} + 3}\, dx+ \mathrm{constant}

Respuesta:

−∫1−19sin⁡2(x)+4sin⁡(2x)+3 dx+constant- \int \frac{1}{- 19 \sin^{2}{\left(x \right)} + 4 \sin{\left(2 x \right)} + 3}\, dx+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                             /        2/x\        /x\\      /       /x\\      /  1      /x\\
 |                                           log|-1 + tan |-| - 8*tan|-||   log|3 + tan|-||   log|- - + tan|-||
 |                1                             \         \2/        \2//      \       \2//      \  3      \2//
 | -------------------------------- dx = C - ---------------------------- + --------------- + -----------------
 |       2                                                16                       16                 16       
 | 19*sin (x) - 8*sin(x)*cos(x) - 3                                                                            
 |                                                                                                             
/
∫1(19sin⁡2(x)−8sin⁡(x)cos⁡(x))−3 dx=C+log⁡(tan⁡(x2)−13)16+log⁡(tan⁡(x2)+3)16−log⁡(tan⁡2(x2)−8tan⁡(x2)−1)16\int \frac{1}{\left(19 \sin^{2}{\left(x \right)} - 8 \sin{\left(x \right)} \cos{\left(x \right)}\right) - 3}\, dx = C + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{3} \right)}}{16} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 3 \right)}}{16} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 8 \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{16}
Simplificar
Respuesta [src] 
     /       2                  \                                                  
  log\1 - tan (1/2) + 8*tan(1/2)/   log(3 + tan(1/2))   log(-1/3 + tan(1/2))   pi*I
- ------------------------------- + ----------------- + -------------------- - ----
                 16                         16                   16             16
−log⁡(−tan⁡2(12)+1+8tan⁡(12))16+log⁡(−13+tan⁡(12))16+log⁡(tan⁡(12)+3)16−iπ16- \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 8 \tan{\left(\frac{1}{2} \right)} \right)}}{16} + \frac{\log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} \right)}}{16} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 3 \right)}}{16} - \frac{i \pi}{16} 
=
= 
     /       2                  \                                                  
  log\1 - tan (1/2) + 8*tan(1/2)/   log(3 + tan(1/2))   log(-1/3 + tan(1/2))   pi*I
- ------------------------------- + ----------------- + -------------------- - ----
                 16                         16                   16             16
−log⁡(−tan⁡2(12)+1+8tan⁡(12))16+log⁡(−13+tan⁡(12))16+log⁡(tan⁡(12)+3)16−iπ16- \frac{\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 8 \tan{\left(\frac{1}{2} \right)} \right)}}{16} + \frac{\log{\left(- \frac{1}{3} + \tan{\left(\frac{1}{2} \right)} \right)}}{16} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 3 \right)}}{16} - \frac{i \pi}{16} 
-log(1 - tan(1/2)^2 + 8*tan(1/2))/16 + log(3 + tan(1/2))/16 + log(-1/3 + tan(1/2))/16 - pi*i/16
Respuesta numérica [src] 
1.50456229563194
1.50456229563194

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

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