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