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Resolución de integrales/ cosx+2/ 4cosx/ 2sinx/ 1/(3-4cosx+2sinx)

Integral de 1/(3-4cosx+2sinx) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |             1              
 |  ----------------------- dx
 |  3 - 4*cos(x) + 2*sin(x)   
 |                            
/                             
0
∫011(3−4cos⁡(x))+2sin⁡(x) dx\int\limits_{0}^{1} \frac{1}{\left(3 - 4 \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}}\, dx 
Integral(1/(3 - 4*cos(x) + 2*sin(x)), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

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

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

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

      Pero la integral

      11log⁡(tan⁡(x2)+27+117)11−11log⁡(tan⁡(x2)−117+27)11\frac{\sqrt{11} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{2}{7} + \frac{\sqrt{11}}{7} \right)}}{11} - \frac{\sqrt{11} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{11}}{7} + \frac{2}{7} \right)}}{11}

    Por lo tanto, el resultado es: −11log⁡(tan⁡(x2)+27+117)11+11log⁡(tan⁡(x2)−117+27)11- \frac{\sqrt{11} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{2}{7} + \frac{\sqrt{11}}{7} \right)}}{11} + \frac{\sqrt{11} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{11}}{7} + \frac{2}{7} \right)}}{11}

  3. Ahora simplificar:

    11(−log⁡(tan⁡(x2)+27+117)+log⁡(tan⁡(x2)−117+27))11\frac{\sqrt{11} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{2}{7} + \frac{\sqrt{11}}{7} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{11}}{7} + \frac{2}{7} \right)}\right)}{11}

  4. Añadimos la constante de integración:

    11(−log⁡(tan⁡(x2)+27+117)+log⁡(tan⁡(x2)−117+27))11+constant\frac{\sqrt{11} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{2}{7} + \frac{\sqrt{11}}{7} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{11}}{7} + \frac{2}{7} \right)}\right)}{11}+ \mathrm{constant}

Respuesta:

11(−log⁡(tan⁡(x2)+27+117)+log⁡(tan⁡(x2)−117+27))11+constant\frac{\sqrt{11} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{2}{7} + \frac{\sqrt{11}}{7} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{11}}{7} + \frac{2}{7} \right)}\right)}{11}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                              /      ____         \             /      ____         \
  /                                   ____    |2   \/ 11       /x\|     ____    |2   \/ 11       /x\|
 |                                  \/ 11 *log|- + ------ + tan|-||   \/ 11 *log|- - ------ + tan|-||
 |            1                               \7     7         \2//             \7     7         \2//
 | ----------------------- dx = C - ------------------------------- + -------------------------------
 | 3 - 4*cos(x) + 2*sin(x)                         11                                11              
 |                                                                                                   
/
∫1(3−4cos⁡(x))+2sin⁡(x) dx=C−11log⁡(tan⁡(x2)+27+117)11+11log⁡(tan⁡(x2)−117+27)11\int \frac{1}{\left(3 - 4 \cos{\left(x \right)}\right) + 2 \sin{\left(x \right)}}\, dx = C - \frac{\sqrt{11} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{2}{7} + \frac{\sqrt{11}}{7} \right)}}{11} + \frac{\sqrt{11} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{11}}{7} + \frac{2}{7} \right)}}{11}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1000010000
Trazar el gráfico f(x)
Respuesta [src] 
         /          /        ____\\             /      ____           \             /      ____\             /      ____           \
    ____ |          |  2   \/ 11 ||     ____    |2   \/ 11            |     ____    |2   \/ 11 |     ____    |2   \/ 11            |
  \/ 11 *|pi*I + log|- - + ------||   \/ 11 *log|- + ------ + tan(1/2)|   \/ 11 *log|- + ------|   \/ 11 *log|- - ------ + tan(1/2)|
         \          \  7     7   //             \7     7              /             \7     7   /             \7     7              /
- --------------------------------- - --------------------------------- + ---------------------- + ---------------------------------
                  11                                  11                            11                             11
11log⁡(−117+27+tan⁡(12))11+11log⁡(27+117)11−11log⁡(27+117+tan⁡(12))11−11(log⁡(−27+117)+iπ)11\frac{\sqrt{11} \log{\left(- \frac{\sqrt{11}}{7} + \frac{2}{7} + \tan{\left(\frac{1}{2} \right)} \right)}}{11} + \frac{\sqrt{11} \log{\left(\frac{2}{7} + \frac{\sqrt{11}}{7} \right)}}{11} - \frac{\sqrt{11} \log{\left(\frac{2}{7} + \frac{\sqrt{11}}{7} + \tan{\left(\frac{1}{2} \right)} \right)}}{11} - \frac{\sqrt{11} \left(\log{\left(- \frac{2}{7} + \frac{\sqrt{11}}{7} \right)} + i \pi\right)}{11} 
=
= 
         /          /        ____\\             /      ____           \             /      ____\             /      ____           \
    ____ |          |  2   \/ 11 ||     ____    |2   \/ 11            |     ____    |2   \/ 11 |     ____    |2   \/ 11            |
  \/ 11 *|pi*I + log|- - + ------||   \/ 11 *log|- + ------ + tan(1/2)|   \/ 11 *log|- + ------|   \/ 11 *log|- - ------ + tan(1/2)|
         \          \  7     7   //             \7     7              /             \7     7   /             \7     7              /
- --------------------------------- - --------------------------------- + ---------------------- + ---------------------------------
                  11                                  11                            11                             11
11log⁡(−117+27+tan⁡(12))11+11log⁡(27+117)11−11log⁡(27+117+tan⁡(12))11−11(log⁡(−27+117)+iπ)11\frac{\sqrt{11} \log{\left(- \frac{\sqrt{11}}{7} + \frac{2}{7} + \tan{\left(\frac{1}{2} \right)} \right)}}{11} + \frac{\sqrt{11} \log{\left(\frac{2}{7} + \frac{\sqrt{11}}{7} \right)}}{11} - \frac{\sqrt{11} \log{\left(\frac{2}{7} + \frac{\sqrt{11}}{7} + \tan{\left(\frac{1}{2} \right)} \right)}}{11} - \frac{\sqrt{11} \left(\log{\left(- \frac{2}{7} + \frac{\sqrt{11}}{7} \right)} + i \pi\right)}{11} 
-sqrt(11)*(pi*i + log(-2/7 + sqrt(11)/7))/11 - sqrt(11)*log(2/7 + sqrt(11)/7 + tan(1/2))/11 + sqrt(11)*log(2/7 + sqrt(11)/7)/11 + sqrt(11)*log(2/7 - sqrt(11)/7 + tan(1/2))/11
Respuesta numérica [src] 
-0.390064732954375
-0.390064732954375

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

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