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Resolución de integrales/ sh(x)/ x/(cosh(x))^2

Integral de x/(cosh(x))^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     x       
 |  -------- dx
 |      2      
 |  cosh (x)   
 |             
/              
0
$$\int\limits_{0}^{1} \frac{x}{\cosh^{2}{\left(x \right)}}\, dx$$ 
Integral(x/cosh(x)^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                    /        2/x\\        /        /x\\          2/x\        2/x\    /        2/x\\           /x\          2/x\    /        /x\\
 |                                  log|1 + tanh |-||   2*log|1 + tanh|-||    x*tanh |-|    tanh |-|*log|1 + tanh |-||   2*x*tanh|-|    2*tanh |-|*log|1 + tanh|-||
 |    x                   x            \         \2//        \        \2//           \2/         \2/    \         \2//           \2/           \2/    \        \2//
 | -------- dx = C - ------------ - ----------------- + ------------------ - ------------ - -------------------------- + ------------ + ---------------------------
 |     2                     2/x\              2/x\                2/x\              2/x\                  2/x\                  2/x\                   2/x\       
 | cosh (x)          1 + tanh |-|      1 + tanh |-|        1 + tanh |-|      1 + tanh |-|          1 + tanh |-|          1 + tanh |-|           1 + tanh |-|       
 |                            \2/               \2/                 \2/               \2/                   \2/                   \2/                    \2/       
/
$$\int \frac{x}{\cosh^{2}{\left(x \right)}}\, dx = C - \frac{x \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 x \tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{x}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{x}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} + 1}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                         2             /        2     \                                               2         /        2     \         2                        
        1            tanh (1/2)     log\1 + tanh (1/2)/   2*log(1 + tanh(1/2))    2*tanh(1/2)     tanh (1/2)*log\1 + tanh (1/2)/   2*tanh (1/2)*log(1 + tanh(1/2))
- -------------- - -------------- - ------------------- + -------------------- + -------------- - ------------------------------ + -------------------------------
          2                2                   2                     2                   2                        2                                 2             
  1 + tanh (1/2)   1 + tanh (1/2)      1 + tanh (1/2)        1 + tanh (1/2)      1 + tanh (1/2)           1 + tanh (1/2)                    1 + tanh (1/2)
$$- \frac{1}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1}$$ 
=
= 
                         2             /        2     \                                               2         /        2     \         2                        
        1            tanh (1/2)     log\1 + tanh (1/2)/   2*log(1 + tanh(1/2))    2*tanh(1/2)     tanh (1/2)*log\1 + tanh (1/2)/   2*tanh (1/2)*log(1 + tanh(1/2))
- -------------- - -------------- - ------------------- + -------------------- + -------------- - ------------------------------ + -------------------------------
          2                2                   2                     2                   2                        2                                 2             
  1 + tanh (1/2)   1 + tanh (1/2)      1 + tanh (1/2)        1 + tanh (1/2)      1 + tanh (1/2)           1 + tanh (1/2)                    1 + tanh (1/2)
$$- \frac{1}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1}$$ 
-1/(1 + tanh(1/2)^2) - tanh(1/2)^2/(1 + tanh(1/2)^2) - log(1 + tanh(1/2)^2)/(1 + tanh(1/2)^2) + 2*log(1 + tanh(1/2))/(1 + tanh(1/2)^2) + 2*tanh(1/2)/(1 + tanh(1/2)^2) - tanh(1/2)^2*log(1 + tanh(1/2)^2)/(1 + tanh(1/2)^2) + 2*tanh(1/2)^2*log(1 + tanh(1/2))/(1 + tanh(1/2)^2)
Respuesta numérica [src] 
0.327813325472738
0.327813325472738

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

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