Sr Examen
Integral de lnx/(x^3+1) dx
Solución
Ha introducido
[src]
oo / | | log(x) | ------ dx | 3 | x + 1 | / 1
$$\int\limits_{1}^{\infty} \frac{\log{\left(x \right)}}{x^{3} + 1}\, dx$$
Integral(log(x)/(x^3 + 1), (x, 1, oo))
Respuesta
[src]
/ -2*pi*I / pi*I\ 2*pi*I / -pi*I \\
| ------- | ----| ------ | ------||
| 2 3 | 3 | 3 | 3 ||
2 | pi 4*e *polylog\2, e / 4*e *polylog\2, e /|
Gamma (2/3)*|- --- + ---------------------------- + -----------------------------|
\ 9 3 3 /
----------------------------------------------------------------------------------
2
9*Gamma (5/3)
$$\frac{\left(- \frac{\pi^{2}}{9} + \frac{4 e^{- \frac{2 i \pi}{3}} \operatorname{Li}_{2}\left(e^{\frac{i \pi}{3}}\right)}{3} + \frac{4 e^{\frac{2 i \pi}{3}} \operatorname{Li}_{2}\left(e^{- \frac{i \pi}{3}}\right)}{3}\right) \Gamma^{2}\left(\frac{2}{3}\right)}{9 \Gamma^{2}\left(\frac{5}{3}\right)}$$
=
=
/ -2*pi*I / pi*I\ 2*pi*I / -pi*I \\
| ------- | ----| ------ | ------||
| 2 3 | 3 | 3 | 3 ||
2 | pi 4*e *polylog\2, e / 4*e *polylog\2, e /|
Gamma (2/3)*|- --- + ---------------------------- + -----------------------------|
\ 9 3 3 /
----------------------------------------------------------------------------------
2
9*Gamma (5/3)
$$\frac{\left(- \frac{\pi^{2}}{9} + \frac{4 e^{- \frac{2 i \pi}{3}} \operatorname{Li}_{2}\left(e^{\frac{i \pi}{3}}\right)}{3} + \frac{4 e^{\frac{2 i \pi}{3}} \operatorname{Li}_{2}\left(e^{- \frac{i \pi}{3}}\right)}{3}\right) \Gamma^{2}\left(\frac{2}{3}\right)}{9 \Gamma^{2}\left(\frac{5}{3}\right)}$$
gamma(2/3)^2*(-pi^2/9 + 4*exp(-2*pi*i/3)*polylog(2, exp(pi*i/3))/3 + 4*exp(2*pi*i/3)*polylog(2, exp(-pi*i/3))/3)/(9*gamma(5/3)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.