Sr Examen
Integral de log10(x+2)/x dx
Solución
Ha introducido
[src]
2 / | | /log(x + 2)\ | |----------| | \ log(10) / | ------------ dx | x | / 6/5
$$\int\limits_{\frac{6}{5}}^{2} \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x + 2 \right)}}{x}\, dx$$
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)
|
/
$$\int \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x + 2 \right)}}{x}\, dx = C + \frac{\begin{cases} \log{\left(2 \right)} \log{\left(x \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{for}\: \left|{x}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x} \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{otherwise} \end{cases}}{\log{\left(10 \right)}}$$
Gráfica
Respuesta
[src]
2
2 pi
log (2) + ---
12 -polylog(2, -3/5) + log(2)*log(6/5)
------------- - -----------------------------------
log(10) log(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)
$$- \frac{\log{\left(\frac{6}{5} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(- \frac{3}{5}\right)}{\log{\left(10 \right)}} + \frac{\log{\left(2 \right)}^{2} + \frac{\pi^{2}}{12}}{\log{\left(10 \right)}}$$
(log(2)^2 + pi^2/12)/log(10) - (-polylog(2, -3/5) + log(2)*log(6/5))/log(10)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.