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Resolución de integrales/ e^(x/4)/ e^(-x/4)/ 1/(e^(x/4)+4e^(-x/4))

Integral de 1/(e^(x/4)+4e^(-x/4)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   4*log(2)                
       /                   
      |                    
      |           1        
      |      ----------- dx
      |       x      -x    
      |       -      ---   
      |       4       4    
      |      E  + 4*E      
      |                    
     /                     
     /  2  \               
4*log|-----|               
     |3 ___|               
     \\/ x /
∫4log⁡(2x3)4log⁡(2)14e(−1)x4+ex4 dx\int\limits_{4 \log{\left(\frac{2}{\sqrt[3]{x}} \right)}}^{4 \log{\left(2 \right)}} \frac{1}{4 e^{\frac{\left(-1\right) x}{4}} + e^{\frac{x}{4}}}\, dx 
Integral(1/(E^(x/4) + 4*E^((-x)/4)), (x, 4*log(2/x^(1/3)), 4*log(2)))
Respuesta (Indefinida) [src] 
                              / x\
  /                           | -|
 |                            | 4|
 |      1                     |e |
 | ----------- dx = C + 2*atan|--|
 |  x      -x                 \2 /
 |  -      ---                    
 |  4       4                     
 | E  + 4*E                       
 |                                
/
∫14e(−1)x4+ex4 dx=C+2atan⁡(ex42)\int \frac{1}{4 e^{\frac{\left(-1\right) x}{4}} + e^{\frac{x}{4}}}\, dx = C + 2 \operatorname{atan}{\left(\frac{e^{\frac{x}{4}}}{2} \right)}
Simplificar
Respuesta [src] 
         / 2                /  2        \\          / 2                         \
- RootSum|z  + 1, i -> i*log|----- + 2*i|| + RootSum\z  + 1, i -> i*log(2 + 2*i)/
         |                  |3 ___      ||                                       
         \                  \\/ x       //
RootSum⁡(z2+1,(i↦ilog⁡(2i+2)))−RootSum⁡(z2+1,(i↦ilog⁡(2i+2x3)))\operatorname{RootSum} {\left(z^{2} + 1, \left( i \mapsto i \log{\left(2 i + 2 \right)} \right)\right)} - \operatorname{RootSum} {\left(z^{2} + 1, \left( i \mapsto i \log{\left(2 i + \frac{2}{\sqrt[3]{x}} \right)} \right)\right)} 
=
= 
         / 2                /  2        \\          / 2                         \
- RootSum|z  + 1, i -> i*log|----- + 2*i|| + RootSum\z  + 1, i -> i*log(2 + 2*i)/
         |                  |3 ___      ||                                       
         \                  \\/ x       //
RootSum⁡(z2+1,(i↦ilog⁡(2i+2)))−RootSum⁡(z2+1,(i↦ilog⁡(2i+2x3)))\operatorname{RootSum} {\left(z^{2} + 1, \left( i \mapsto i \log{\left(2 i + 2 \right)} \right)\right)} - \operatorname{RootSum} {\left(z^{2} + 1, \left( i \mapsto i \log{\left(2 i + \frac{2}{\sqrt[3]{x}} \right)} \right)\right)} 
-RootSum(_z^2 + 1, Lambda(_i, _i*log(2/x^(1/3) + 2*_i))) + RootSum(_z^2 + 1, Lambda(_i, _i*log(2 + 2*_i)))

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

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