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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |         sin(x)         
 |  ------------------- dx
 |  2 - sin(x) + cos(x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right) + \cos{\left(x \right)}}\, dx$$ 
Integral(sin(x)/(2 - sin(x) + cos(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                /       2/x\\          /       2/x\        /x\\         /        /x   pi\       /            ___    /x\\\
 |                              log|1 + tan |-||       log|3 + tan |-| - 2*tan|-||         |        |- - --|       |    ___   \/ 2 *tan|-|||
 |        sin(x)                   \        \2//   x      \        \2/        \2//     ___ |        |2   2 |       |  \/ 2             \2/||
 | ------------------- dx = C + ---------------- - - - --------------------------- + \/ 2 *|pi*floor|------| + atan|- ----- + ------------||
 | 2 - sin(x) + cos(x)                 2           2                2                      \        \  pi  /       \    2          2      //
 |                                                                                                                                          
/
$$\int \frac{\sin{\left(x \right)}}{\left(2 - \sin{\left(x \right)}\right) + \cos{\left(x \right)}}\, dx = C - \frac{x}{2} + \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                  /       2     \      /       2                  \         /          /  ___     ___         \\         /          /  ___\\
  1   log(3)   log\1 + tan (1/2)/   log\3 + tan (1/2) - 2*tan(1/2)/     ___ |          |\/ 2    \/ 2 *tan(1/2)||     ___ |          |\/ 2 ||
- - + ------ + ------------------ - ------------------------------- + \/ 2 *|-pi - atan|----- - --------------|| - \/ 2 *|-pi - atan|-----||
  2     2              2                           2                        \          \  2           2       //         \          \  2  //
$$\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)}\right) - \frac{1}{2} - \frac{\log{\left(- 2 \tan{\left(\frac{1}{2} \right)} + \tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} - \sqrt{2} \left(- \pi - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right)$$ 
=
= 
                  /       2     \      /       2                  \         /          /  ___     ___         \\         /          /  ___\\
  1   log(3)   log\1 + tan (1/2)/   log\3 + tan (1/2) - 2*tan(1/2)/     ___ |          |\/ 2    \/ 2 *tan(1/2)||     ___ |          |\/ 2 ||
- - + ------ + ------------------ - ------------------------------- + \/ 2 *|-pi - atan|----- - --------------|| - \/ 2 *|-pi - atan|-----||
  2     2              2                           2                        \          \  2           2       //         \          \  2  //
$$\sqrt{2} \left(- \pi - \operatorname{atan}{\left(- \frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)}\right) - \frac{1}{2} - \frac{\log{\left(- 2 \tan{\left(\frac{1}{2} \right)} + \tan^{2}{\left(\frac{1}{2} \right)} + 3 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} - \sqrt{2} \left(- \pi - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right)$$ 
-1/2 + log(3)/2 + log(1 + tan(1/2)^2)/2 - log(3 + tan(1/2)^2 - 2*tan(1/2))/2 + sqrt(2)*(-pi - atan(sqrt(2)/2 - sqrt(2)*tan(1/2)/2)) - sqrt(2)*(-pi - atan(sqrt(2)/2))
Respuesta numérica [src] 
0.215727333211525
0.215727333211525

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

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