Sr Examen
Integral de lnx/(X^2+a^2) dx
Solución
Ha introducido
[src]
oo / | | log(x) | ------- dx | 2 2 | x + a | / 0
$$\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}}{a^{2} + x^{2}}\, dx$$
Integral(log(x)/(x^2 + a^2), (x, 0, oo))
Respuesta
[src]
/ / pi*I\ / 3*pi*I\ | | ----| | ------| | | 2 | | 2 | / 3*pi*I\ / pi*I\ | | e | | e | | ------| | ----| | - 2*I*a*polylog|2, -----| + 2*I*a*polylog|2, -------| | 2 | | 2 | | \ a / \ a / I*polylog\2, a*e / I*polylog\2, a*e / |- ----------------------------------------------------- + ----------------------- - --------------------- for 2*|arg(a)| < pi | 2 2*a 2*a | 4*a | < oo | / | | | | log(x) | | ------- dx otherwise | | 2 2 | | a + x | | | / | 0 \
$$\begin{cases} - \frac{i \operatorname{Li}_{2}\left(a e^{\frac{i \pi}{2}}\right)}{2 a} + \frac{i \operatorname{Li}_{2}\left(a e^{\frac{3 i \pi}{2}}\right)}{2 a} - \frac{- 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{i \pi}{2}}}{a}\right) + 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{3 i \pi}{2}}}{a}\right)}{4 a^{2}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}}{a^{2} + x^{2}}\, dx & \text{otherwise} \end{cases}$$
=
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/ / pi*I\ / 3*pi*I\ | | ----| | ------| | | 2 | | 2 | / 3*pi*I\ / pi*I\ | | e | | e | | ------| | ----| | - 2*I*a*polylog|2, -----| + 2*I*a*polylog|2, -------| | 2 | | 2 | | \ a / \ a / I*polylog\2, a*e / I*polylog\2, a*e / |- ----------------------------------------------------- + ----------------------- - --------------------- for 2*|arg(a)| < pi | 2 2*a 2*a | 4*a | < oo | / | | | | log(x) | | ------- dx otherwise | | 2 2 | | a + x | | | / | 0 \
$$\begin{cases} - \frac{i \operatorname{Li}_{2}\left(a e^{\frac{i \pi}{2}}\right)}{2 a} + \frac{i \operatorname{Li}_{2}\left(a e^{\frac{3 i \pi}{2}}\right)}{2 a} - \frac{- 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{i \pi}{2}}}{a}\right) + 2 i a \operatorname{Li}_{2}\left(\frac{e^{\frac{3 i \pi}{2}}}{a}\right)}{4 a^{2}} & \text{for}\: 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{\log{\left(x \right)}}{a^{2} + x^{2}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-(-2*i*a*polylog(2, exp_polar(pi*i/2)/a) + 2*i*a*polylog(2, exp_polar(3*pi*i/2)/a))/(4*a^2) + i*polylog(2, a*exp_polar(3*pi*i/2))/(2*a) - i*polylog(2, a*exp_polar(pi*i/2))/(2*a), 2*Abs(arg(a)) < pi), (Integral(log(x)/(a^2 + x^2), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.