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Resolución de integrales/ sh(x)/ sinh(x)/ (cosh(x))^3÷(1+sinh(x))

Integral de (cosh(x))^3÷(1+sinh(x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |        3       
 |    cosh (x)    
 |  ----------- dx
 |  1 + sinh(x)   
 |                
/                 
0
$$\int\limits_{0}^{1} \frac{\cosh^{3}{\left(x \right)}}{\sinh{\left(x \right)} + 1}\, dx$$ 
Integral(cosh(x)^3/(1 + sinh(x)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                                                                                                                                                                                                                                                                             
 |                               /        /x\\                    /x\                                                     2/x\                        3/x\               /         2/x\         /x\\                  2/x\               2/x\    /         2/x\         /x\\         4/x\    /        /x\\                  4/x\               4/x\    /         2/x\         /x\\         2/x\    /        /x\\
 |       3                  4*log|1 + tanh|-||              2*tanh|-|                                               2*tanh |-|                  2*tanh |-|          2*log|-1 + tanh |-| - 2*tanh|-||          4*x*tanh |-|         4*tanh |-|*log|-1 + tanh |-| - 2*tanh|-||   4*tanh |-|*log|1 + tanh|-||          2*x*tanh |-|         2*tanh |-|*log|-1 + tanh |-| - 2*tanh|-||   8*tanh |-|*log|1 + tanh|-||
 |   cosh (x)                    \        \2//                    \2/                      2*x                             \2/                         \2/               \          \2/         \2//                   \2/                \2/    \          \2/         \2//          \2/    \        \2//                   \2/                \2/    \          \2/         \2//          \2/    \        \2//
 | ----------- dx = C - ------------------------- - ------------------------- + ------------------------- + ------------------------- + ------------------------- + -------------------------------- - ------------------------- - ----------------------------------------- - --------------------------- + ------------------------- + ----------------------------------------- + ---------------------------
 | 1 + sinh(x)                  4/x\         2/x\           4/x\         2/x\           4/x\         2/x\           4/x\         2/x\           4/x\         2/x\              4/x\         2/x\               4/x\         2/x\                   4/x\         2/x\                    4/x\         2/x\            4/x\         2/x\                   4/x\         2/x\                    4/x\         2/x\ 
 |                      1 + tanh |-| - 2*tanh |-|   1 + tanh |-| - 2*tanh |-|   1 + tanh |-| - 2*tanh |-|   1 + tanh |-| - 2*tanh |-|   1 + tanh |-| - 2*tanh |-|      1 + tanh |-| - 2*tanh |-|       1 + tanh |-| - 2*tanh |-|           1 + tanh |-| - 2*tanh |-|            1 + tanh |-| - 2*tanh |-|    1 + tanh |-| - 2*tanh |-|           1 + tanh |-| - 2*tanh |-|            1 + tanh |-| - 2*tanh |-| 
/                                \2/          \2/            \2/          \2/            \2/          \2/            \2/          \2/            \2/          \2/               \2/          \2/                \2/          \2/                    \2/          \2/                     \2/          \2/             \2/          \2/                    \2/          \2/                     \2/          \2/
$$\int \frac{\cosh^{3}{\left(x \right)}}{\sinh{\left(x \right)} + 1}\, dx = C + \frac{2 x \tanh^{4}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 x \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 x}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)} \tanh^{4}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{8 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 2 \tanh{\left(\frac{x}{2} \right)} - 1 \right)} \tanh^{4}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 \log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 2 \tanh{\left(\frac{x}{2} \right)} - 1 \right)} \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} - 2 \tanh{\left(\frac{x}{2} \right)} - 1 \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tanh^{3}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \tanh{\left(\frac{x}{2} \right)}}{\tanh^{4}{\left(\frac{x}{2} \right)} - 2 \tanh^{2}{\left(\frac{x}{2} \right)} + 1}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                        2                                                               3                               4                  /          /        2                   \\         2      /          /        2                   \\         4                                 4      /          /        2                   \\         2                        
              2                      4*log(1 + tanh(1/2))                         2*tanh (1/2)                    2*tanh(1/2)                     2*tanh (1/2)                    2*tanh (1/2)           2*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)//   4*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)//   4*tanh (1/2)*log(1 + tanh(1/2))   2*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)//   8*tanh (1/2)*log(1 + tanh(1/2))
----------------------------- - ----------------------------- - 2*pi*I - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- + -------------------------------------------- - ------------------------------------------------------- - ------------------------------- + ------------------------------------------------------- + -------------------------------
        4              2                4              2                         4              2                4              2                4              2                4              2                       4              2                                     4              2                              4              2                              4              2                              4              2      
1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)            1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)          1 + tanh (1/2) - 2*tanh (1/2)                        1 + tanh (1/2) - 2*tanh (1/2)                 1 + tanh (1/2) - 2*tanh (1/2)                 1 + tanh (1/2) - 2*tanh (1/2)                 1 + tanh (1/2) - 2*tanh (1/2)
$$- \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{3}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{8 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - 2 i \pi - \frac{4 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1}$$ 
=
= 
                                                                                        2                                                               3                               4                  /          /        2                   \\         2      /          /        2                   \\         4                                 4      /          /        2                   \\         2                        
              2                      4*log(1 + tanh(1/2))                         2*tanh (1/2)                    2*tanh(1/2)                     2*tanh (1/2)                    2*tanh (1/2)           2*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)//   4*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)//   4*tanh (1/2)*log(1 + tanh(1/2))   2*tanh (1/2)*\pi*I + log\1 - tanh (1/2) + 2*tanh(1/2)//   8*tanh (1/2)*log(1 + tanh(1/2))
----------------------------- - ----------------------------- - 2*pi*I - ----------------------------- - ----------------------------- + ----------------------------- + ----------------------------- + -------------------------------------------- - ------------------------------------------------------- - ------------------------------- + ------------------------------------------------------- + -------------------------------
        4              2                4              2                         4              2                4              2                4              2                4              2                       4              2                                     4              2                              4              2                              4              2                              4              2      
1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)            1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)   1 + tanh (1/2) - 2*tanh (1/2)          1 + tanh (1/2) - 2*tanh (1/2)                        1 + tanh (1/2) - 2*tanh (1/2)                 1 + tanh (1/2) - 2*tanh (1/2)                 1 + tanh (1/2) - 2*tanh (1/2)                 1 + tanh (1/2) - 2*tanh (1/2)
$$- \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{2 \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - \frac{4 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh^{3}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{8 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} - 2 i \pi - \frac{4 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{2}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tanh^{4}{\left(\frac{1}{2} \right)}}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \left(\log{\left(- \tanh^{2}{\left(\frac{1}{2} \right)} + 2 \tanh{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{- 2 \tanh^{2}{\left(\frac{1}{2} \right)} + \tanh^{4}{\left(\frac{1}{2} \right)} + 1}$$ 
2/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 4*log(1 + tanh(1/2))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 2*pi*i - 2*tanh(1/2)^2/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 2*tanh(1/2)/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*tanh(1/2)^3/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*tanh(1/2)^4/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*(pi*i + log(1 - tanh(1/2)^2 + 2*tanh(1/2)))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 4*tanh(1/2)^2*(pi*i + log(1 - tanh(1/2)^2 + 2*tanh(1/2)))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) - 4*tanh(1/2)^4*log(1 + tanh(1/2))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 2*tanh(1/2)^4*(pi*i + log(1 - tanh(1/2)^2 + 2*tanh(1/2)))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2) + 8*tanh(1/2)^2*log(1 + tanh(1/2))/(1 + tanh(1/2)^4 - 2*tanh(1/2)^2)
Respuesta numérica [src] 
1.06959005530163
1.06959005530163

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

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