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Resolución de integrales/ sh(x)/ ch(x)/ 1/ch(x)*((sh(x))^2/2+C)

Integral de 1/ch(x)*((sh(x))^2/2+C) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |      2          
 |  sinh (x)       
 |  -------- + c   
 |     2           
 |  ------------ dx
 |    cosh(x)      
 |                 
/                  
0
∫01c+sinh⁡2(x)2cosh⁡(x) dx\int\limits_{0}^{1} \frac{c + \frac{\sinh^{2}{\left(x \right)}}{2}}{\cosh{\left(x \right)}}\, dx 
Integral((sinh(x)^2/2 + c)/cosh(x), (x, 0, 1))
Solución detallada

  1. Hay varias maneras de calcular esta integral.

    Método #1

    1. Vuelva a escribir el integrando:

      c+sinh⁡2(x)2cosh⁡(x)=2c+sinh⁡2(x)2cosh⁡(x)\frac{c + \frac{\sinh^{2}{\left(x \right)}}{2}}{\cosh{\left(x \right)}} = \frac{2 c + \sinh^{2}{\left(x \right)}}{2 \cosh{\left(x \right)}}

    2. La integral del producto de una función por una constante es la constante por la integral de esta función:

      ∫2c+sinh⁡2(x)2cosh⁡(x) dx=∫2c+sinh⁡2(x)cosh⁡(x) dx2\int \frac{2 c + \sinh^{2}{\left(x \right)}}{2 \cosh{\left(x \right)}}\, dx = \frac{\int \frac{2 c + \sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)}}\, dx}{2}

      1. Vuelva a escribir el integrando:

        2c+sinh⁡2(x)cosh⁡(x)=2ccosh⁡(x)+sinh⁡2(x)cosh⁡(x)\frac{2 c + \sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)}} = \frac{2 c}{\cosh{\left(x \right)}} + \frac{\sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)}}

      2. Integramos término a término:

        1. La integral del producto de una función por una constante es la constante por la integral de esta función:

          ∫2ccosh⁡(x) dx=2c∫1cosh⁡(x) dx\int \frac{2 c}{\cosh{\left(x \right)}}\, dx = 2 c \int \frac{1}{\cosh{\left(x \right)}}\, dx

          1. No puedo encontrar los pasos en la búsqueda de esta integral.

            Pero la integral

            2atan⁡(tanh⁡(x2))2 \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}

          Por lo tanto, el resultado es: 4catan⁡(tanh⁡(x2))4 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}

        1. No puedo encontrar los pasos en la búsqueda de esta integral.

          Pero la integral

          −2tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−2tanh⁡(x2)tanh⁡2(x2)−1+2atan⁡(tanh⁡(x2))tanh⁡2(x2)−1- \frac{2 \tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{2 \tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{2 \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}

        El resultado es: 4catan⁡(tanh⁡(x2))−2tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−2tanh⁡(x2)tanh⁡2(x2)−1+2atan⁡(tanh⁡(x2))tanh⁡2(x2)−14 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} - \frac{2 \tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{2 \tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{2 \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}

      Por lo tanto, el resultado es: 2catan⁡(tanh⁡(x2))−tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−tanh⁡(x2)tanh⁡2(x2)−1+atan⁡(tanh⁡(x2))tanh⁡2(x2)−12 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} - \frac{\tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}

    Método #2

    1. Vuelva a escribir el integrando:

      c+sinh⁡2(x)2cosh⁡(x)=ccosh⁡(x)+sinh⁡2(x)2cosh⁡(x)\frac{c + \frac{\sinh^{2}{\left(x \right)}}{2}}{\cosh{\left(x \right)}} = \frac{c}{\cosh{\left(x \right)}} + \frac{\sinh^{2}{\left(x \right)}}{2 \cosh{\left(x \right)}}

    2. Integramos término a término:

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫ccosh⁡(x) dx=c∫1cosh⁡(x) dx\int \frac{c}{\cosh{\left(x \right)}}\, dx = c \int \frac{1}{\cosh{\left(x \right)}}\, dx

        1. No puedo encontrar los pasos en la búsqueda de esta integral.

          Pero la integral

          2atan⁡(tanh⁡(x2))2 \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}

        Por lo tanto, el resultado es: 2catan⁡(tanh⁡(x2))2 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫sinh⁡2(x)2cosh⁡(x) dx=∫sinh⁡2(x)cosh⁡(x) dx2\int \frac{\sinh^{2}{\left(x \right)}}{2 \cosh{\left(x \right)}}\, dx = \frac{\int \frac{\sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)}}\, dx}{2}

        1. No puedo encontrar los pasos en la búsqueda de esta integral.

          Pero la integral

          −2tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−2tanh⁡(x2)tanh⁡2(x2)−1+2atan⁡(tanh⁡(x2))tanh⁡2(x2)−1- \frac{2 \tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{2 \tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{2 \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}

        Por lo tanto, el resultado es: −tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−tanh⁡(x2)tanh⁡2(x2)−1+atan⁡(tanh⁡(x2))tanh⁡2(x2)−1- \frac{\tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}

      El resultado es: 2catan⁡(tanh⁡(x2))−tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−tanh⁡(x2)tanh⁡2(x2)−1+atan⁡(tanh⁡(x2))tanh⁡2(x2)−12 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} - \frac{\tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}

  2. Ahora simplificar:

    2catan⁡(tanh⁡(x2))+sinh⁡(x)2−atan⁡(tanh⁡(x2))2 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} + \frac{\sinh{\left(x \right)}}{2} - \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}

  3. Añadimos la constante de integración:

    2catan⁡(tanh⁡(x2))+sinh⁡(x)2−atan⁡(tanh⁡(x2))+constant2 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} + \frac{\sinh{\left(x \right)}}{2} - \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}+ \mathrm{constant}

Respuesta:

2catan⁡(tanh⁡(x2))+sinh⁡(x)2−atan⁡(tanh⁡(x2))+constant2 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} + \frac{\sinh{\left(x \right)}}{2} - \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                
 |                                                                                                 
 |     2                                                                                           
 | sinh (x)                  /    /x\\          /x\                              2/x\     /    /x\\
 | -------- + c          atan|tanh|-||      tanh|-|                          tanh |-|*atan|tanh|-||
 |    2                      \    \2//          \2/              /    /x\\        \2/     \    \2//
 | ------------ dx = C + ------------- - ------------- + 2*c*atan|tanh|-|| - ----------------------
 |   cosh(x)                      2/x\            2/x\           \    \2//                2/x\     
 |                       -1 + tanh |-|   -1 + tanh |-|                           -1 + tanh |-|     
/                                  \2/             \2/                                     \2/
∫c+sinh⁡2(x)2cosh⁡(x) dx=C+2catan⁡(tanh⁡(x2))−tanh⁡2(x2)atan⁡(tanh⁡(x2))tanh⁡2(x2)−1−tanh⁡(x2)tanh⁡2(x2)−1+atan⁡(tanh⁡(x2))tanh⁡2(x2)−1\int \frac{c + \frac{\sinh^{2}{\left(x \right)}}{2}}{\cosh{\left(x \right)}}\, dx = C + 2 c \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)} - \frac{\tanh^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\tanh{\left(\frac{x}{2} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{x}{2} \right)} \right)}}{\tanh^{2}{\left(\frac{x}{2} \right)} - 1}
Simplificar
Respuesta [src] 
                                        2                                                      2                     
atan(tanh(1/2))      tanh(1/2)      tanh (1/2)*atan(tanh(1/2))   2*c*atan(tanh(1/2))   2*c*tanh (1/2)*atan(tanh(1/2))
--------------- - --------------- - -------------------------- - ------------------- + ------------------------------
         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)
2ctanh⁡2(12)atan⁡(tanh⁡(12))−1+tanh⁡2(12)−2catan⁡(tanh⁡(12))−1+tanh⁡2(12)+atan⁡(tanh⁡(12))−1+tanh⁡2(12)−tanh⁡2(12)atan⁡(tanh⁡(12))−1+tanh⁡2(12)−tanh⁡(12)−1+tanh⁡2(12)\frac{2 c \tanh^{2}{\left(\frac{1}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} - \frac{2 c \operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} - \frac{\tanh^{2}{\left(\frac{1}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} - \frac{\tanh{\left(\frac{1}{2} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} 
=
= 
                                        2                                                      2                     
atan(tanh(1/2))      tanh(1/2)      tanh (1/2)*atan(tanh(1/2))   2*c*atan(tanh(1/2))   2*c*tanh (1/2)*atan(tanh(1/2))
--------------- - --------------- - -------------------------- - ------------------- + ------------------------------
         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)
2ctanh⁡2(12)atan⁡(tanh⁡(12))−1+tanh⁡2(12)−2catan⁡(tanh⁡(12))−1+tanh⁡2(12)+atan⁡(tanh⁡(12))−1+tanh⁡2(12)−tanh⁡2(12)atan⁡(tanh⁡(12))−1+tanh⁡2(12)−tanh⁡(12)−1+tanh⁡2(12)\frac{2 c \tanh^{2}{\left(\frac{1}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} - \frac{2 c \operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} + \frac{\operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} - \frac{\tanh^{2}{\left(\frac{1}{2} \right)} \operatorname{atan}{\left(\tanh{\left(\frac{1}{2} \right)} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} - \frac{\tanh{\left(\frac{1}{2} \right)}}{-1 + \tanh^{2}{\left(\frac{1}{2} \right)}} 
atan(tanh(1/2))/(-1 + tanh(1/2)^2) - tanh(1/2)/(-1 + tanh(1/2)^2) - tanh(1/2)^2*atan(tanh(1/2))/(-1 + tanh(1/2)^2) - 2*c*atan(tanh(1/2))/(-1 + tanh(1/2)^2) + 2*c*tanh(1/2)^2*atan(tanh(1/2))/(-1 + tanh(1/2)^2)

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

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