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Resolución de integrales/ sin(x)/ x/(1+sin(x))

Integral de x/(1+sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |      x        
 |  ---------- dx
 |  1 + sin(x)   
 |               
/                
0
$$\int\limits_{0}^{1} \frac{x}{\sin{\left(x \right)} + 1}\, dx$$ 
Integral(x/(1 + sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                    /       2/x\\        /       /x\\         /x\       /       2/x\\    /x\        /       /x\\    /x\
 |                                  log|1 + tan |-||   2*log|1 + tan|-||    x*tan|-|    log|1 + tan |-||*tan|-|   2*log|1 + tan|-||*tan|-|
 |     x                   x           \        \2//        \       \2//         \2/       \        \2//    \2/        \       \2//    \2/
 | ---------- dx = C - ---------- - ---------------- + ----------------- + ---------- - ----------------------- + ------------------------
 | 1 + sin(x)                 /x\             /x\                 /x\             /x\                 /x\                       /x\       
 |                     1 + tan|-|      1 + tan|-|          1 + tan|-|      1 + tan|-|          1 + tan|-|                1 + tan|-|       
/                             \2/             \2/                 \2/             \2/                 \2/                       \2/
$$\int \frac{x}{\sin{\left(x \right)} + 1}\, dx = C + \frac{x \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{x}{\tan{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan{\left(\frac{x}{2} \right)} + 1}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                   /       2     \                            /       2     \                                        
       1           tan(1/2)     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 + tan(1/2)   1 + tan(1/2)      1 + tan(1/2)          1 + tan(1/2)              1 + tan(1/2)                  1 + tan(1/2)
$$- \frac{1}{\tan{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} + \frac{\tan{\left(\frac{1}{2} \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan{\left(\frac{1}{2} \right)} + 1}$$ 
=
= 
                                   /       2     \                            /       2     \                                        
       1           tan(1/2)     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 + tan(1/2)   1 + tan(1/2)      1 + tan(1/2)          1 + tan(1/2)              1 + tan(1/2)                  1 + tan(1/2)
$$- \frac{1}{\tan{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} + \frac{\tan{\left(\frac{1}{2} \right)}}{\tan{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{\tan{\left(\frac{1}{2} \right)} + 1}$$ 
-1/(1 + tan(1/2)) + tan(1/2)/(1 + tan(1/2)) - log(1 + tan(1/2)^2)/(1 + tan(1/2)) + 2*log(1 + tan(1/2))/(1 + tan(1/2)) - log(1 + tan(1/2)^2)*tan(1/2)/(1 + tan(1/2)) + 2*log(1 + tan(1/2))*tan(1/2)/(1 + tan(1/2))
Respuesta numérica [src] 
0.317156707471479
0.317156707471479

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

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