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Resolución de integrales/ 1+sinx/ 1/(cosx(1+sinx))

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |  cos(x)*(1 + sin(x))   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{1}{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}\, dx$$ 
Integral(1/(cos(x)*(1 + sin(x))), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                    /       /x\\               /        /x\\                    /x\              2/x\    /       /x\\       2/x\    /        /x\\        /        /x\\    /x\        /       /x\\    /x\
 |                                  log|1 + tan|-||            log|-1 + tan|-||               2*tan|-|           tan |-|*log|1 + tan|-||    tan |-|*log|-1 + tan|-||   2*log|-1 + tan|-||*tan|-|   2*log|1 + tan|-||*tan|-|
 |          1                          \       \2//               \        \2//                    \2/               \2/    \       \2//        \2/    \        \2//        \        \2//    \2/        \       \2//    \2/
 | ------------------- dx = C + ------------------------ - ------------------------ + ------------------------ + ------------------------ - ------------------------ - ------------------------- + ------------------------
 | cos(x)*(1 + sin(x))                   2/x\        /x\            2/x\        /x\            2/x\        /x\            2/x\        /x\            2/x\        /x\             2/x\        /x\            2/x\        /x\
 |                              2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|    2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|
/                                         \2/        \2/             \2/        \2/             \2/        \2/             \2/        \2/             \2/        \2/              \2/        \2/             \2/        \2/
$$\int \frac{1}{\left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}\, dx = C - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                                       2                              2                                                                                                       
     log(1 + tan(1/2))         pi*I     pi*I + log(1 - tan(1/2))              2*tan(1/2)            tan (1/2)*log(1 + tan(1/2))    tan (1/2)*(pi*I + log(1 - tan(1/2)))   2*(pi*I + log(1 - tan(1/2)))*tan(1/2)   2*log(1 + tan(1/2))*tan(1/2)
---------------------------- + ---- - ---------------------------- + ---------------------------- + ---------------------------- - ------------------------------------ - ------------------------------------- + ----------------------------
         2                      2              2                              2                              2                                  2                                       2                                  2                  
2 + 2*tan (1/2) + 4*tan(1/2)          2 + 2*tan (1/2) + 4*tan(1/2)   2 + 2*tan (1/2) + 4*tan(1/2)   2 + 2*tan (1/2) + 4*tan(1/2)       2 + 2*tan (1/2) + 4*tan(1/2)            2 + 2*tan (1/2) + 4*tan(1/2)       2 + 2*tan (1/2) + 4*tan(1/2)
$$\frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \tan{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} - \frac{2 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} - \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{i \pi}{2}$$ 
=
= 
                                                                                                       2                              2                                                                                                       
     log(1 + tan(1/2))         pi*I     pi*I + log(1 - tan(1/2))              2*tan(1/2)            tan (1/2)*log(1 + tan(1/2))    tan (1/2)*(pi*I + log(1 - tan(1/2)))   2*(pi*I + log(1 - tan(1/2)))*tan(1/2)   2*log(1 + tan(1/2))*tan(1/2)
---------------------------- + ---- - ---------------------------- + ---------------------------- + ---------------------------- - ------------------------------------ - ------------------------------------- + ----------------------------
         2                      2              2                              2                              2                                  2                                       2                                  2                  
2 + 2*tan (1/2) + 4*tan(1/2)          2 + 2*tan (1/2) + 4*tan(1/2)   2 + 2*tan (1/2) + 4*tan(1/2)   2 + 2*tan (1/2) + 4*tan(1/2)       2 + 2*tan (1/2) + 4*tan(1/2)            2 + 2*tan (1/2) + 4*tan(1/2)       2 + 2*tan (1/2) + 4*tan(1/2)
$$\frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{2 \tan{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} - \frac{2 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} - \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{2}{\left(\frac{1}{2} \right)} + 2 + 4 \tan{\left(\frac{1}{2} \right)}} + \frac{i \pi}{2}$$ 
log(1 + tan(1/2))/(2 + 2*tan(1/2)^2 + 4*tan(1/2)) + pi*i/2 - (pi*i + log(1 - tan(1/2)))/(2 + 2*tan(1/2)^2 + 4*tan(1/2)) + 2*tan(1/2)/(2 + 2*tan(1/2)^2 + 4*tan(1/2)) + tan(1/2)^2*log(1 + tan(1/2))/(2 + 2*tan(1/2)^2 + 4*tan(1/2)) - tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(2 + 2*tan(1/2)^2 + 4*tan(1/2)) - 2*(pi*i + log(1 - tan(1/2)))*tan(1/2)/(2 + 2*tan(1/2)^2 + 4*tan(1/2)) + 2*log(1 + tan(1/2))*tan(1/2)/(2 + 2*tan(1/2)^2 + 4*tan(1/2))
Respuesta numérica [src] 
0.841573522848869
0.841573522848869

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

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