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Resolución de integrales/ (shx)/ 1/((chx)^2+(shx)^2)

Integral de 1/((chx)^2+(shx)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |           1            
 |  ------------------- dx
 |      2          2      
 |  cosh (x) + sinh (x)   
 |                        
/                         
0
$$\int\limits_{0}^{1} \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx$$ 
Integral(1/(cosh(x)^2 + sinh(x)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                              /        /x\     \                                                  /        /x\     \
                                              |    tanh|-|     |               _____________    _____________     |    tanh|-|     |
                                              |        \2/     |              /         ___    /         ___      |        \2/     |
                                          atan|----------------|            \/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|
  /                                           |   _____________|                                                  |   _____________|
 |                                            |  /         ___ |                                                  |  /         ___ |
 |          1                                 \\/  3 - 2*\/ 2  /                                                  \\/  3 + 2*\/ 2  /
 | ------------------- dx = C + ----------------------------------------- - --------------------------------------------------------
 |     2          2                _____________            _____________             _____________            _____________        
 | cosh (x) + sinh (x)            /         ___      ___   /         ___             /         ___      ___   /         ___         
 |                              \/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2            \/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2          
/
$$\int \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx = C + \frac{\operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
                                               _____________    _____________                       
              /   tanh(1/2)    \              /         ___    /         ___      /   tanh(1/2)    \
          atan|----------------|            \/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|
              |   _____________|                                                  |   _____________|
              |  /         ___ |                                                  |  /         ___ |
              \\/  3 - 2*\/ 2  /                                                  \\/  3 + 2*\/ 2  /
----------------------------------------- - --------------------------------------------------------
   _____________            _____________             _____________            _____________        
  /         ___      ___   /         ___             /         ___      ___   /         ___         
\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2            \/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2
$$- \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{\operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$ 
=
= 
                                               _____________    _____________                       
              /   tanh(1/2)    \              /         ___    /         ___      /   tanh(1/2)    \
          atan|----------------|            \/  3 - 2*\/ 2  *\/  3 + 2*\/ 2  *atan|----------------|
              |   _____________|                                                  |   _____________|
              |  /         ___ |                                                  |  /         ___ |
              \\/  3 - 2*\/ 2  /                                                  \\/  3 + 2*\/ 2  /
----------------------------------------- - --------------------------------------------------------
   _____________            _____________             _____________            _____________        
  /         ___      ___   /         ___             /         ___      ___   /         ___         
\/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2            \/  3 - 2*\/ 2   + \/ 2 *\/  3 - 2*\/ 2
$$- \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{\operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{\sqrt{3 - 2 \sqrt{2}} + \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$ 
atan(tanh(1/2)/sqrt(3 - 2*sqrt(2)))/(sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2))) - sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*atan(tanh(1/2)/sqrt(3 + 2*sqrt(2)))/(sqrt(3 - 2*sqrt(2)) + sqrt(2)*sqrt(3 - 2*sqrt(2)))
Respuesta numérica [src] 
0.650880168023008
0.650880168023008

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

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