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