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