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