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Resolución de integrales/ sin(x)/ sin(x)/(5+sin(x))

Integral de sin(x)/(5+sin(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |    sin(x)     
 |  ---------- dx
 |  5 + sin(x)   
 |               
/                
0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 5}\, dx$$ 
Integral(sin(x)/(5 + sin(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                   /        /x   pi\       /            ___    /x\\\
                                   |        |- - --|       |  ___   5*\/ 6 *tan|-|||
  /                            ___ |        |2   2 |       |\/ 6               \2/||
 |                         5*\/ 6 *|pi*floor|------| + atan|----- + --------------||
 |   sin(x)                        \        \  pi  /       \  12          12      //
 | ---------- dx = C + x - ---------------------------------------------------------
 | 5 + sin(x)                                          6                            
 |                                                                                  
/
$$\int \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 5}\, dx = C + x - \frac{5 \sqrt{6} \left(\operatorname{atan}{\left(\frac{5 \sqrt{6} \tan{\left(\frac{x}{2} \right)}}{12} + \frac{\sqrt{6}}{12} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{6}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
            /          /  ___       ___         \\           /          /  ___\\
        ___ |          |\/ 6    5*\/ 6 *tan(1/2)||       ___ |          |\/ 6 ||
    5*\/ 6 *|-pi + atan|----- + ----------------||   5*\/ 6 *|-pi + atan|-----||
            \          \  12           12       //           \          \  12 //
1 - ---------------------------------------------- + ---------------------------
                          6                                       6
$$\frac{5 \sqrt{6} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)}\right)}{6} + 1 - \frac{5 \sqrt{6} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{6}}{12} + \frac{5 \sqrt{6} \tan{\left(\frac{1}{2} \right)}}{12} \right)}\right)}{6}$$ 
=
= 
            /          /  ___       ___         \\           /          /  ___\\
        ___ |          |\/ 6    5*\/ 6 *tan(1/2)||       ___ |          |\/ 6 ||
    5*\/ 6 *|-pi + atan|----- + ----------------||   5*\/ 6 *|-pi + atan|-----||
            \          \  12           12       //           \          \  12 //
1 - ---------------------------------------------- + ---------------------------
                          6                                       6
$$\frac{5 \sqrt{6} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)}\right)}{6} + 1 - \frac{5 \sqrt{6} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{6}}{12} + \frac{5 \sqrt{6} \tan{\left(\frac{1}{2} \right)}}{12} \right)}\right)}{6}$$ 
1 - 5*sqrt(6)*(-pi + atan(sqrt(6)/12 + 5*sqrt(6)*tan(1/2)/12))/6 + 5*sqrt(6)*(-pi + atan(sqrt(6)/12))/6
Respuesta numérica [src] 
0.0822904284713761
0.0822904284713761

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

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