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Resolución de integrales/ cos(x)/ cosx// sin(x)/ cosx/(2+cos(x)+sin(x))

Integral de cosx/(2+cos(x)+sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |         cos(x)         
 |  ------------------- dx
 |  2 + cos(x) + sin(x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{\cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 2\right) + \sin{\left(x \right)}}\, dx$$ 
Integral(cos(x)/(2 + cos(x) + sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                    /       2/x\        /x\\      /       2/x\\         /        /x   pi\       /          ___    /x\\\
 |                                  log|3 + tan |-| + 2*tan|-||   log|1 + tan |-||         |        |- - --|       |  ___   \/ 2 *tan|-|||
 |        cos(x)                x      \        \2/        \2//      \        \2//     ___ |        |2   2 |       |\/ 2             \2/||
 | ------------------- dx = C + - + --------------------------- - ---------------- - \/ 2 *|pi*floor|------| + atan|----- + ------------||
 | 2 + cos(x) + sin(x)          2                2                       2                 \        \  pi  /       \  2          2      //
 |                                                                                                                                        
/
$$\int \frac{\cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 2\right) + \sin{\left(x \right)}}\, dx = C + \frac{x}{2} - \sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right) - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
       /       2                  \               /       2     \         /          /  ___\\         /          /  ___     ___         \\
1   log\3 + tan (1/2) + 2*tan(1/2)/   log(3)   log\1 + tan (1/2)/     ___ |          |\/ 2 ||     ___ |          |\/ 2    \/ 2 *tan(1/2)||
- + ------------------------------- - ------ - ------------------ + \/ 2 *|-pi + atan|-----|| - \/ 2 *|-pi + atan|----- + --------------||
2                  2                    2              2                  \          \  2  //         \          \  2           2       //
$$\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right) - \frac{\log{\left(3 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{1}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{2} - \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)}\right)$$ 
=
= 
       /       2                  \               /       2     \         /          /  ___\\         /          /  ___     ___         \\
1   log\3 + tan (1/2) + 2*tan(1/2)/   log(3)   log\1 + tan (1/2)/     ___ |          |\/ 2 ||     ___ |          |\/ 2    \/ 2 *tan(1/2)||
- + ------------------------------- - ------ - ------------------ + \/ 2 *|-pi + atan|-----|| - \/ 2 *|-pi + atan|----- + --------------||
2                  2                    2              2                  \          \  2  //         \          \  2           2       //
$$\sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}\right) - \frac{\log{\left(3 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{1}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 2 \tan{\left(\frac{1}{2} \right)} + 3 \right)}}{2} - \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{1}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)}\right)$$ 
1/2 + log(3 + tan(1/2)^2 + 2*tan(1/2))/2 - log(3)/2 - log(1 + tan(1/2)^2)/2 + sqrt(2)*(-pi + atan(sqrt(2)/2)) - sqrt(2)*(-pi + atan(sqrt(2)/2 + sqrt(2)*tan(1/2)/2))
Respuesta numérica [src] 
0.256537031237107
0.256537031237107

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

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