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