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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                          
  /                          
 |                           
 |            1              
 |  ---------------------- dx
 |                       2   
 |  (1 + cos(x) + sin(x))    
 |                           
/                            
0
$$\int\limits_{0}^{1} \frac{1}{\left(\left(\cos{\left(x \right)} + 1\right) + \sin{\left(x \right)}\right)^{2}}\, dx$$ 
Integral(1/((1 + cos(x) + sin(x))^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                    2/x\           /       /x\\        /       /x\\    /x\
 |                                                  tan |-|      2*log|1 + tan|-||   2*log|1 + tan|-||*tan|-|
 |           1                          3               \2/           \       \2//        \       \2//    \2/
 | ---------------------- dx = C - ------------ + ------------ - ----------------- - ------------------------
 |                      2                   /x\            /x\               /x\                    /x\      
 | (1 + cos(x) + sin(x))           2 + 2*tan|-|   2 + 2*tan|-|      2 + 2*tan|-|           2 + 2*tan|-|      
 |                                          \2/            \2/               \2/                    \2/      
/
$$\int \frac{1}{\left(\left(\cos{\left(x \right)} + 1\right) + \sin{\left(x \right)}\right)^{2}}\, dx = C - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan{\left(\frac{x}{2} \right)} + 2} - \frac{3}{2 \tan{\left(\frac{x}{2} \right)} + 2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                          2                                                             
3         3            tan (1/2)      2*log(1 + tan(1/2))   2*log(1 + tan(1/2))*tan(1/2)
- - -------------- + -------------- - ------------------- - ----------------------------
2   2 + 2*tan(1/2)   2 + 2*tan(1/2)      2 + 2*tan(1/2)            2 + 2*tan(1/2)
$$- \frac{3}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{3}{2}$$ 
=
= 
                          2                                                             
3         3            tan (1/2)      2*log(1 + tan(1/2))   2*log(1 + tan(1/2))*tan(1/2)
- - -------------- + -------------- - ------------------- - ----------------------------
2   2 + 2*tan(1/2)   2 + 2*tan(1/2)      2 + 2*tan(1/2)            2 + 2*tan(1/2)
$$- \frac{3}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan{\left(\frac{1}{2} \right)} + 2} + \frac{3}{2}$$ 
3/2 - 3/(2 + 2*tan(1/2)) + tan(1/2)^2/(2 + 2*tan(1/2)) - 2*log(1 + 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] 
0.190580657716409
0.190580657716409

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

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