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Resolución de integrales/ (x+2)/ log10/ log10(x+2)/x

Integral de log10(x+2)/x dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  2                
  /                
 |                 
 |  /log(x + 2)\   
 |  |----------|   
 |  \ log(10)  /   
 |  ------------ dx
 |       x         
 |                 
/                  
6/5
∫6521log⁡(10)log⁡(x+2)x dx\int\limits_{\frac{6}{5}}^{2} \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x + 2 \right)}}{x}\, dx 
Integral((log(x + 2)/log(10))/x, (x, 6/5, 2))
Respuesta (Indefinida) [src] 
                         /                                   /      pi*I\                                                       
                         |                                   |   x*e    |                                                       
                         |                          - polylog|2, -------| + log(2)*log(x)                            for |x| < 1
                         |                                   \      2   /                                                       
                         |                                                                                                      
                         |                                   /      pi*I\                                                       
                         |                                   |   x*e    |             /1\                                 1     
                         <                          - polylog|2, -------| - log(2)*log|-|                            for --- < 1
                         |                                   \      2   /             \x/                                |x|    
                         |                                                                                                      
  /                      |         /      pi*I\                                                                                 
 |                       |         |   x*e    |           __0, 2 /1, 1       |  \           __2, 0 /      1, 1 |  \             
 | /log(x + 2)\          |- polylog|2, -------| + log(2)*/__     |           | x| - log(2)*/__     |           | x|   otherwise 
 | |----------|          |         \      2   /          \_|2, 2 \      0, 0 |  /          \_|2, 2 \0, 0       |  /             
 | \ log(10)  /          \                                                                                                      
 | ------------ dx = C + -------------------------------------------------------------------------------------------------------
 |      x                                                                log(10)                                                
 |                                                                                                                              
/
∫1log⁡(10)log⁡(x+2)x dx=C+{log⁡(2)log⁡(x)−Li⁡2(xeiπ2)for ∣x∣<1−log⁡(2)log⁡(1x)−Li⁡2(xeiπ2)for 1∣x∣<1−G2,22,0(1,10,0|x)log⁡(2)+G2,20,2(1,10,0|x)log⁡(2)−Li⁡2(xeiπ2)otherwiselog⁡(10)\int \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x + 2 \right)}}{x}\, dx = C + \frac{\begin{cases} \log{\left(2 \right)} \log{\left(x \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{for}\: \left|{x}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x} \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{otherwise} \end{cases}}{\log{\left(10 \right)}}
Simplificar
Gráfica
2.001.201.301.401.501.601.701.801.900.01.0
Trazar el gráfico f(x)
Respuesta [src] 
            2                                      
   2      pi                                       
log (2) + ---                                      
           12   -polylog(2, -3/5) + log(2)*log(6/5)
------------- - -----------------------------------
   log(10)                    log(10)
−log⁡(65)log⁡(2)−Li⁡2(−35)log⁡(10)+log⁡(2)2+π212log⁡(10)- \frac{\log{\left(\frac{6}{5} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(- \frac{3}{5}\right)}{\log{\left(10 \right)}} + \frac{\log{\left(2 \right)}^{2} + \frac{\pi^{2}}{12}}{\log{\left(10 \right)}} 
=
= 
            2                                      
   2      pi                                       
log (2) + ---                                      
           12   -polylog(2, -3/5) + log(2)*log(6/5)
------------- - -----------------------------------
   log(10)                    log(10)
−log⁡(65)log⁡(2)−Li⁡2(−35)log⁡(10)+log⁡(2)2+π212log⁡(10)- \frac{\log{\left(\frac{6}{5} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(- \frac{3}{5}\right)}{\log{\left(10 \right)}} + \frac{\log{\left(2 \right)}^{2} + \frac{\pi^{2}}{12}}{\log{\left(10 \right)}} 
(log(2)^2 + pi^2/12)/log(10) - (-polylog(2, -3/5) + log(2)*log(6/5))/log(10)
Respuesta numérica [src] 
0.281612697929638
0.281612697929638

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

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