Sr Examen
Integral de ln^6(8x+16)/x+2 dx
Solución
Ha introducido
[src]
1 / | | / 6 \ | |log (8*x + 16) | | |-------------- + 2| dx | \ x / | / 0
$$\int\limits_{0}^{1} \left(2 + \frac{\log{\left(8 x + 16 \right)}^{6}}{x}\right)\, dx$$
Integral(log(8*x + 16)^6/x + 2, (x, 0, 1))
Respuesta (Indefinida)
[src]
// / pi*I\ \
|| | x*e | |
|| - polylog|2, -------| + log(2)*log(x) for |x| < 1|
/ / / \ / / / || \ 2 / | /
| | | | | | | || | |
| / 6 \ | | 5 | | 4 | 3 | 2 || / pi*I\ | | 6
| |log (8*x + 16) | | | log (2 + x) | 2 | log (2 + x) 3 | log (2 + x) 6 4 | log (2 + x) 5 || | x*e | /1\ 1 | | log (2 + x)
| |-------------- + 2| dx = C + 2*x + 18*| | ----------- dx|*log(2) + 135*log (2)* | ----------- dx + 540*log (2)* | ----------- dx + 729*log (2)*log(x) + 1215*log (2)* | ----------- dx + 1458*log (2)*|< - polylog|2, -------| - log(2)*log|-| for --- < 1| + | ----------- dx
| \ x / | | x | | x | x | x || \ 2 / \x/ |x| | | x
| | | | | | | || | |
/ \/ / / / / || / pi*I\ | /
|| | x*e | __0, 2 /1, 1 | \ __2, 0 / 1, 1 | \ |
||- polylog|2, -------| + log(2)*/__ | | x| - log(2)*/__ | | x| otherwise |
|| \ 2 / \_|2, 2 \ 0, 0 | / \_|2, 2 \0, 0 | / |
\\ /
$$\int \left(2 + \frac{\log{\left(8 x + 16 \right)}^{6}}{x}\right)\, dx = C + 2 x + 1458 \left(\begin{cases} \log{\left(2 \right)} \log{\left(x \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{for}\: \left|{x}\right| < 1 \\- \log{\left(2 \right)} \log{\left(\frac{1}{x} \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\- {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} \log{\left(2 \right)} + {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} \log{\left(2 \right)} - \operatorname{Li}_{2}\left(\frac{x e^{i \pi}}{2}\right) & \text{otherwise} \end{cases}\right) \log{\left(2 \right)}^{5} + 729 \log{\left(2 \right)}^{6} \log{\left(x \right)} + 1215 \log{\left(2 \right)}^{4} \int \frac{\log{\left(x + 2 \right)}^{2}}{x}\, dx + 540 \log{\left(2 \right)}^{3} \int \frac{\log{\left(x + 2 \right)}^{3}}{x}\, dx + 135 \log{\left(2 \right)}^{2} \int \frac{\log{\left(x + 2 \right)}^{4}}{x}\, dx + 18 \log{\left(2 \right)} \int \frac{\log{\left(x + 2 \right)}^{5}}{x}\, dx + \int \frac{\log{\left(x + 2 \right)}^{6}}{x}\, dx$$
Respuesta
[src]
1 / | | 6 | log (16 + 8*x) + 2*x | -------------------- dx | x | / 0
$$\int\limits_{0}^{1} \frac{2 x + \log{\left(8 x + 16 \right)}^{6}}{x}\, dx$$
=
=
1 / | | 6 | log (16 + 8*x) + 2*x | -------------------- dx | x | / 0
$$\int\limits_{0}^{1} \frac{2 x + \log{\left(8 x + 16 \right)}^{6}}{x}\, dx$$
Integral((log(16 + 8*x)^6 + 2*x)/x, (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.