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Resolución de integrales/ 2cosx/ x+sinx/ 1/(6-2cosx+sinx)

Integral de 1/(6-2cosx+sinx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                         
  /                         
 |                          
 |            1             
 |  --------------------- dx
 |  6 - 2*cos(x) + sin(x)   
 |                          
/                           
0
∫011(6−2cos⁡(x))+sin⁡(x) dx\int\limits_{0}^{1} \frac{1}{\left(6 - 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)}}\, dx 
Integral(1/(6 - 2*cos(x) + sin(x)), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    1(6−2cos⁡(x))+sin⁡(x)=−1−sin⁡(x)+2cos⁡(x)−6\frac{1}{\left(6 - 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)}} = - \frac{1}{- \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 6}

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

    ∫(−1−sin⁡(x)+2cos⁡(x)−6) dx=−∫1−sin⁡(x)+2cos⁡(x)−6 dx\int \left(- \frac{1}{- \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 6}\right)\, dx = - \int \frac{1}{- \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 6}\, dx

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

      Pero la integral

      −231(atan⁡(831tan⁡(x2)31+3131)+π⌊x2−π2π⌋)31- \frac{2 \sqrt{31} \left(\operatorname{atan}{\left(\frac{8 \sqrt{31} \tan{\left(\frac{x}{2} \right)}}{31} + \frac{\sqrt{31}}{31} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{31}

    Por lo tanto, el resultado es: 231(atan⁡(831tan⁡(x2)31+3131)+π⌊x2−π2π⌋)31\frac{2 \sqrt{31} \left(\operatorname{atan}{\left(\frac{8 \sqrt{31} \tan{\left(\frac{x}{2} \right)}}{31} + \frac{\sqrt{31}}{31} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{31}

  3. Ahora simplificar:

    231(atan⁡(31(8tan⁡(x2)+1)31)+π⌊x2π−12⌋)31\frac{2 \sqrt{31} \left(\operatorname{atan}{\left(\frac{\sqrt{31} \left(8 \tan{\left(\frac{x}{2} \right)} + 1\right)}{31} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{31}

  4. Añadimos la constante de integración:

    231(atan⁡(31(8tan⁡(x2)+1)31)+π⌊x2π−12⌋)31+constant\frac{2 \sqrt{31} \left(\operatorname{atan}{\left(\frac{\sqrt{31} \left(8 \tan{\left(\frac{x}{2} \right)} + 1\right)}{31} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{31}+ \mathrm{constant}

Respuesta:

231(atan⁡(31(8tan⁡(x2)+1)31)+π⌊x2π−12⌋)31+constant\frac{2 \sqrt{31} \left(\operatorname{atan}{\left(\frac{\sqrt{31} \left(8 \tan{\left(\frac{x}{2} \right)} + 1\right)}{31} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{31}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                           /        /x   pi\       /             ____    /x\\\
                                           |        |- - --|       |  ____   8*\/ 31 *tan|-|||
  /                                   ____ |        |2   2 |       |\/ 31                \2/||
 |                                2*\/ 31 *|pi*floor|------| + atan|------ + ---------------||
 |           1                             \        \  pi  /       \  31            31      //
 | --------------------- dx = C + ------------------------------------------------------------
 | 6 - 2*cos(x) + sin(x)                                       31                             
 |                                                                                            
/
∫1(6−2cos⁡(x))+sin⁡(x) dx=C+231(atan⁡(831tan⁡(x2)31+3131)+π⌊x2−π2π⌋)31\int \frac{1}{\left(6 - 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)}}\, dx = C + \frac{2 \sqrt{31} \left(\operatorname{atan}{\left(\frac{8 \sqrt{31} \tan{\left(\frac{x}{2} \right)}}{31} + \frac{\sqrt{31}}{31} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{31}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.901-2
Trazar el gráfico f(x)
Respuesta [src] 
           /          /  ____\\            /          /  ____       ____         \\
      ____ |          |\/ 31 ||       ____ |          |\/ 31    8*\/ 31 *tan(1/2)||
  2*\/ 31 *|-pi + atan|------||   2*\/ 31 *|-pi + atan|------ + -----------------||
           \          \  31  //            \          \  31             31       //
- ----------------------------- + -------------------------------------------------
                31                                        31
231(−π+atan⁡(3131+831tan⁡(12)31))31−231(−π+atan⁡(3131))31\frac{2 \sqrt{31} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{31}}{31} + \frac{8 \sqrt{31} \tan{\left(\frac{1}{2} \right)}}{31} \right)}\right)}{31} - \frac{2 \sqrt{31} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{31}}{31} \right)}\right)}{31} 
=
= 
           /          /  ____\\            /          /  ____       ____         \\
      ____ |          |\/ 31 ||       ____ |          |\/ 31    8*\/ 31 *tan(1/2)||
  2*\/ 31 *|-pi + atan|------||   2*\/ 31 *|-pi + atan|------ + -----------------||
           \          \  31  //            \          \  31             31       //
- ----------------------------- + -------------------------------------------------
                31                                        31
231(−π+atan⁡(3131+831tan⁡(12)31))31−231(−π+atan⁡(3131))31\frac{2 \sqrt{31} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{31}}{31} + \frac{8 \sqrt{31} \tan{\left(\frac{1}{2} \right)}}{31} \right)}\right)}{31} - \frac{2 \sqrt{31} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{31}}{31} \right)}\right)}{31} 
-2*sqrt(31)*(-pi + atan(sqrt(31)/31))/31 + 2*sqrt(31)*(-pi + atan(sqrt(31)/31 + 8*sqrt(31)*tan(1/2)/31))/31
Respuesta numérica [src] 
0.211807716513145
0.211807716513145

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

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