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Resolución de integrales/ sin^2/ 1+cosx/ x+sinx/ sin^2xdx/(1+cosx+sinx)^2

Integral de sin^2xdx/(1+cosx+sinx)^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                          
  /                          
 |                           
 |            2              
 |         sin (x)           
 |  ---------------------- dx
 |                       2   
 |  (1 + cos(x) + sin(x))    
 |                           
/                            
0
∫01sin⁡2(x)((cos⁡(x)+1)+sin⁡(x))2 dx\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{\left(\left(\cos{\left(x \right)} + 1\right) + \sin{\left(x \right)}\right)^{2}}\, dx 
Integral(sin(x)^2/(1 + cos(x) + sin(x))^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                        
 |                                                   /       2/x\\        /       /x\\      /       2/x\\    /x\        /       /x\\    /x\
 |           2                                    log|1 + tan |-||   2*log|1 + tan|-||   log|1 + tan |-||*tan|-|   2*log|1 + tan|-||*tan|-|
 |        sin (x)                       2            \        \2//        \       \2//      \        \2//    \2/        \       \2//    \2/
 | ---------------------- dx = C - ------------ + ---------------- - ----------------- + ----------------------- - ------------------------
 |                      2                   /x\              /x\                 /x\                    /x\                       /x\      
 | (1 + cos(x) + sin(x))           2 + 2*tan|-|     2 + 2*tan|-|        2 + 2*tan|-|           2 + 2*tan|-|              2 + 2*tan|-|      
 |                                          \2/              \2/                 \2/                    \2/                       \2/      
/
∫sin⁡2(x)((cos⁡(x)+1)+sin⁡(x))2 dx=C−2log⁡(tan⁡(x2)+1)tan⁡(x2)2tan⁡(x2)+2−2log⁡(tan⁡(x2)+1)2tan⁡(x2)+2+log⁡(tan⁡2(x2)+1)tan⁡(x2)2tan⁡(x2)+2+log⁡(tan⁡2(x2)+1)2tan⁡(x2)+2−22tan⁡(x2)+2\int \frac{\sin^{2}{\left(x \right)}}{\left(\left(\cos{\left(x \right)} + 1\right) + \sin{\left(x \right)}\right)^{2}}\, dx = C - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} - \frac{2}{2 \tan{\left(\frac{x}{2} \right)} + 2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.901-2
Trazar el gráfico f(x)
Respuesta [src] 
                        /       2     \                            /       2     \                                        
          2          log\1 + tan (1/2)/   2*log(1 + tan(1/2))   log\1 + tan (1/2)/*tan(1/2)   2*log(1 + tan(1/2))*tan(1/2)
1 - -------------- + ------------------ - ------------------- + --------------------------- - ----------------------------
    2 + 2*tan(1/2)     2 + 2*tan(1/2)        2 + 2*tan(1/2)            2 + 2*tan(1/2)                2 + 2*tan(1/2)
−22tan⁡(12)+2−2log⁡(tan⁡(12)+1)2tan⁡(12)+2−2log⁡(tan⁡(12)+1)tan⁡(12)2tan⁡(12)+2+log⁡(tan⁡2(12)+1)tan⁡(12)2tan⁡(12)+2+log⁡(tan⁡2(12)+1)2tan⁡(12)+2+1- \frac{2}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + 1 
=
= 
                        /       2     \                            /       2     \                                        
          2          log\1 + tan (1/2)/   2*log(1 + tan(1/2))   log\1 + tan (1/2)/*tan(1/2)   2*log(1 + tan(1/2))*tan(1/2)
1 - -------------- + ------------------ - ------------------- + --------------------------- - ----------------------------
    2 + 2*tan(1/2)     2 + 2*tan(1/2)        2 + 2*tan(1/2)            2 + 2*tan(1/2)                2 + 2*tan(1/2)
−22tan⁡(12)+2−2log⁡(tan⁡(12)+1)2tan⁡(12)+2−2log⁡(tan⁡(12)+1)tan⁡(12)2tan⁡(12)+2+log⁡(tan⁡2(12)+1)tan⁡(12)2tan⁡(12)+2+log⁡(tan⁡2(12)+1)2tan⁡(12)+2+1- \frac{2}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + 1 
1 - 2/(2 + 2*tan(1/2)) + log(1 + tan(1/2)^2)/(2 + 2*tan(1/2)) - 2*log(1 + tan(1/2))/(2 + 2*tan(1/2)) + log(1 + tan(1/2)^2)*tan(1/2)/(2 + 2*tan(1/2)) - 2*log(1 + tan(1/2))*tan(1/2)/(2 + 2*tan(1/2))
Respuesta numérica [src] 
0.0480136532382369
0.0480136532382369

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

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