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Resolución de integrales/ (7+3(sinx/cosx))/(sinx+2cosx)^2

Integral de (7+3(sinx/cosx))/(sinx+2cosx)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                        
  /                        
 |                         
 |            sin(x)       
 |      7 + 3*------       
 |            cos(x)       
 |  -------------------- dx
 |                     2   
 |  (sin(x) + 2*cos(x))    
 |                         
/                          
0
∫013sin⁡(x)cos⁡(x)+7(sin⁡(x)+2cos⁡(x))2 dx\int\limits_{0}^{1} \frac{3 \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 7}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx 
Integral((7 + 3*(sin(x)/cos(x)))/(sin(x) + 2*cos(x))^2, (x, 0, 1))
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9005
Trazar el gráfico f(x)
Respuesta [src] 
                                    /          /       2                \\                                                                        2                                        2                            /          /       2                \\                 2      /          /       2                \\                                                                        
             tan(1/2)             6*\pi*I + log\1 - tan (1/2) + tan(1/2)//    6*(pi*I + log(1 - tan(1/2)))        6*log(1 + tan(1/2))        6*tan (1/2)*(pi*I + log(1 - tan(1/2)))   6*tan (1/2)*log(1 + tan(1/2))   6*\pi*I + log\1 - tan (1/2) + tan(1/2)//*tan(1/2)   6*tan (1/2)*\pi*I + log\1 - tan (1/2) + tan(1/2)//   6*(pi*I + log(1 - tan(1/2)))*tan(1/2)    6*log(1 + tan(1/2))*tan(1/2)
- ----------------------------- - ---------------------------------------- + ----------------------------- + ----------------------------- - -------------------------------------- - ----------------------------- - ------------------------------------------------- + -------------------------------------------------- + ------------------------------------- + -----------------------------
                         2                                    2                                     2                               2                                   2                                    2                                         2                                                   2                                              2                                   2     
  -2 - 2*tan(1/2) + 2*tan (1/2)        -2 - 2*tan(1/2) + 2*tan (1/2)         -2 - 2*tan(1/2) + 2*tan (1/2)   -2 - 2*tan(1/2) + 2*tan (1/2)       -2 - 2*tan(1/2) + 2*tan (1/2)        -2 - 2*tan(1/2) + 2*tan (1/2)             -2 - 2*tan(1/2) + 2*tan (1/2)                       -2 - 2*tan(1/2) + 2*tan (1/2)                  -2 - 2*tan(1/2) + 2*tan (1/2)       -2 - 2*tan(1/2) + 2*tan (1/2)
6log⁡(tan⁡(12)+1)−2−2tan⁡(12)+2tan⁡2(12)+6log⁡(tan⁡(12)+1)tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)−tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)−6log⁡(tan⁡(12)+1)tan⁡2(12)−2−2tan⁡(12)+2tan⁡2(12)+6(log⁡(1−tan⁡(12))+iπ)−2−2tan⁡(12)+2tan⁡2(12)+6(log⁡(1−tan⁡(12))+iπ)tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)+6(log⁡(−tan⁡2(12)+tan⁡(12)+1)+iπ)tan⁡2(12)−2−2tan⁡(12)+2tan⁡2(12)−6(log⁡(1−tan⁡(12))+iπ)tan⁡2(12)−2−2tan⁡(12)+2tan⁡2(12)−6(log⁡(−tan⁡2(12)+tan⁡(12)+1)+iπ)tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)−6(log⁡(−tan⁡2(12)+tan⁡(12)+1)+iπ)−2−2tan⁡(12)+2tan⁡2(12)\frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} 
=
= 
                                    /          /       2                \\                                                                        2                                        2                            /          /       2                \\                 2      /          /       2                \\                                                                        
             tan(1/2)             6*\pi*I + log\1 - tan (1/2) + tan(1/2)//    6*(pi*I + log(1 - tan(1/2)))        6*log(1 + tan(1/2))        6*tan (1/2)*(pi*I + log(1 - tan(1/2)))   6*tan (1/2)*log(1 + tan(1/2))   6*\pi*I + log\1 - tan (1/2) + tan(1/2)//*tan(1/2)   6*tan (1/2)*\pi*I + log\1 - tan (1/2) + tan(1/2)//   6*(pi*I + log(1 - tan(1/2)))*tan(1/2)    6*log(1 + tan(1/2))*tan(1/2)
- ----------------------------- - ---------------------------------------- + ----------------------------- + ----------------------------- - -------------------------------------- - ----------------------------- - ------------------------------------------------- + -------------------------------------------------- + ------------------------------------- + -----------------------------
                         2                                    2                                     2                               2                                   2                                    2                                         2                                                   2                                              2                                   2     
  -2 - 2*tan(1/2) + 2*tan (1/2)        -2 - 2*tan(1/2) + 2*tan (1/2)         -2 - 2*tan(1/2) + 2*tan (1/2)   -2 - 2*tan(1/2) + 2*tan (1/2)       -2 - 2*tan(1/2) + 2*tan (1/2)        -2 - 2*tan(1/2) + 2*tan (1/2)             -2 - 2*tan(1/2) + 2*tan (1/2)                       -2 - 2*tan(1/2) + 2*tan (1/2)                  -2 - 2*tan(1/2) + 2*tan (1/2)       -2 - 2*tan(1/2) + 2*tan (1/2)
6log⁡(tan⁡(12)+1)−2−2tan⁡(12)+2tan⁡2(12)+6log⁡(tan⁡(12)+1)tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)−tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)−6log⁡(tan⁡(12)+1)tan⁡2(12)−2−2tan⁡(12)+2tan⁡2(12)+6(log⁡(1−tan⁡(12))+iπ)−2−2tan⁡(12)+2tan⁡2(12)+6(log⁡(1−tan⁡(12))+iπ)tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)+6(log⁡(−tan⁡2(12)+tan⁡(12)+1)+iπ)tan⁡2(12)−2−2tan⁡(12)+2tan⁡2(12)−6(log⁡(1−tan⁡(12))+iπ)tan⁡2(12)−2−2tan⁡(12)+2tan⁡2(12)−6(log⁡(−tan⁡2(12)+tan⁡(12)+1)+iπ)tan⁡(12)−2−2tan⁡(12)+2tan⁡2(12)−6(log⁡(−tan⁡2(12)+tan⁡(12)+1)+iπ)−2−2tan⁡(12)+2tan⁡2(12)\frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} 
-tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*(pi*i + log(1 - tan(1/2)^2 + tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*(pi*i + log(1 - tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*log(1 + tan(1/2))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*tan(1/2)^2*log(1 + tan(1/2))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*(pi*i + log(1 - tan(1/2)^2 + tan(1/2)))*tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*tan(1/2)^2*(pi*i + log(1 - tan(1/2)^2 + tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*(pi*i + log(1 - tan(1/2)))*tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*log(1 + tan(1/2))*tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2)
Respuesta numérica [src] 
1.94655122867851
1.94655122867851

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

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