Sr Examen
Integral de log10(x)*(x+10)^(-1) dx
Solución
Ha introducido
[src]
25 / | | / log(x)\ | |-------| | \log(10)/ | --------- dx | x + 10 | / 0
$$\int\limits_{0}^{25} \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x \right)}}{x + 10}\, dx$$
Integral((log(x)/log(10))/(x + 10), (x, 0, 25))
Respuesta (Indefinida)
[src]
/ / x \
| - polylog|2, 1 + --| + log(10)*log(10 + x) + 3*pi*I*log(10 + x) for |10 + x| < 1
| \ 10/
|
| / x \ / 1 \ / 1 \ 1
< - polylog|2, 1 + --| - log(10)*log|------| - 3*pi*I*log|------| for -------- < 1
/ | \ 10/ \10 + x/ \10 + x/ |10 + x|
| |
| / log(x)\ | / x \ __0, 2 /1, 1 | \ __2, 0 / 1, 1 | \ __2, 0 / 1, 1 | \ __0, 2 /1, 1 | \
| |-------| |- polylog|2, 1 + --| + log(10)*/__ | | 10 + x| - log(10)*/__ | | 10 + x| - 3*pi*I*/__ | | 10 + x| + 3*pi*I*/__ | | 10 + x| otherwise
| \log(10)/ \ \ 10/ \_|2, 2 \ 0, 0 | / \_|2, 2 \0, 0 | / \_|2, 2 \0, 0 | / \_|2, 2 \ 0, 0 | /
| --------- dx = C + -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| x + 10 log(10)
|
/
$$\int \frac{\frac{1}{\log{\left(10 \right)}} \log{\left(x \right)}}{x + 10}\, dx = C + \frac{\begin{cases} \log{\left(10 \right)} \log{\left(x + 10 \right)} + 3 i \pi \log{\left(x + 10 \right)} - \operatorname{Li}_{2}\left(\frac{x}{10} + 1\right) & \text{for}\: \left|{x + 10}\right| < 1 \\- \log{\left(10 \right)} \log{\left(\frac{1}{x + 10} \right)} - 3 i \pi \log{\left(\frac{1}{x + 10} \right)} - \operatorname{Li}_{2}\left(\frac{x}{10} + 1\right) & \text{for}\: \frac{1}{\left|{x + 10}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x + 10} \right)} \log{\left(10 \right)} - 3 i \pi {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x + 10} \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x + 10} \right)} \log{\left(10 \right)} + 3 i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x + 10} \right)} - \operatorname{Li}_{2}\left(\frac{x}{10} + 1\right) & \text{otherwise} \end{cases}}{\log{\left(10 \right)}}$$
Gráfica
Respuesta
[src]
2
2 pi
log (10) - --- + 3*pi*I*log(10)
-polylog(2, 7/2) + log(10)*log(35) + 3*pi*I*log(35) 6
--------------------------------------------------- - -------------------------------
log(10) log(10)
$$- \frac{- \frac{\pi^{2}}{6} + \log{\left(10 \right)}^{2} + 3 i \pi \log{\left(10 \right)}}{\log{\left(10 \right)}} + \frac{\log{\left(10 \right)} \log{\left(35 \right)} - \operatorname{Li}_{2}\left(\frac{7}{2}\right) + 3 i \pi \log{\left(35 \right)}}{\log{\left(10 \right)}}$$
=
=
2
2 pi
log (10) - --- + 3*pi*I*log(10)
-polylog(2, 7/2) + log(10)*log(35) + 3*pi*I*log(35) 6
--------------------------------------------------- - -------------------------------
log(10) log(10)
$$- \frac{- \frac{\pi^{2}}{6} + \log{\left(10 \right)}^{2} + 3 i \pi \log{\left(10 \right)}}{\log{\left(10 \right)}} + \frac{\log{\left(10 \right)} \log{\left(35 \right)} - \operatorname{Li}_{2}\left(\frac{7}{2}\right) + 3 i \pi \log{\left(35 \right)}}{\log{\left(10 \right)}}$$
(-polylog(2, 7/2) + log(10)*log(35) + 3*pi*i*log(35))/log(10) - (log(10)^2 - pi^2/6 + 3*pi*i*log(10))/log(10)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.