Sr Examen
Integral de ln(1+x)/((x)^(1/3)*(4+x^2)) dx
Solución
Ha introducido
[src]
oo / | | log(1 + x) | -------------- dx | 3 ___ / 2\ | \/ x *\4 + x / | / 0
$$\int\limits_{0}^{\infty} \frac{\log{\left(x + 1 \right)}}{\sqrt[3]{x} \left(x^{2} + 4\right)}\, dx$$
Integral(log(1 + x)/((x^(1/3)*(4 + x^2))), (x, 0, oo))
Respuesta
[src]
/ / -2*pi*I / pi*I\ 2*pi*I / 5*pi*I\\ \
| | ------- | ----| ------ | ------|| |
| | 2/3 / 2/3 pi*I\ 2/3 3 | 2/3 3 | 2/3 3 | 2/3 3 || |
| 3 ___ 2 | 2 *log\1 - 2 *e / 2 *e *log\1 - 2 *e / 2 *e *log\1 - 2 *e /| |
| 2*\/ 2 *Gamma (2/3)*|- ------------------------ - --------------------------------- - ----------------------------------|*Gamma(-1/6)*Gamma(1/3)*Gamma(7/6)|
2/3 | \ 6 6 6 / |
2 *|-pi*Gamma(-1/3)*Gamma(4/3)*log(5) + 2*pi*atan(2)*Gamma(1/6)*Gamma(5/6) + -----------------------------------------------------------------------------------------------------------------------------------------------------------|
\ Gamma(5/3) /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16*pi
$$\frac{2^{\frac{2}{3}} \left(\frac{2 \sqrt[3]{2} \left(- \frac{2^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} \log{\left(- 2^{\frac{2}{3}} e^{\frac{5 i \pi}{3}} + 1 \right)}}{6} - \frac{2^{\frac{2}{3}} \log{\left(1 - 2^{\frac{2}{3}} e^{i \pi} \right)}}{6} - \frac{2^{\frac{2}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(1 - 2^{\frac{2}{3}} e^{\frac{i \pi}{3}} \right)}}{6}\right) \Gamma\left(- \frac{1}{6}\right) \Gamma\left(\frac{1}{3}\right) \Gamma^{2}\left(\frac{2}{3}\right) \Gamma\left(\frac{7}{6}\right)}{\Gamma\left(\frac{5}{3}\right)} - \pi \log{\left(5 \right)} \Gamma\left(- \frac{1}{3}\right) \Gamma\left(\frac{4}{3}\right) + 2 \pi \operatorname{atan}{\left(2 \right)} \Gamma\left(\frac{1}{6}\right) \Gamma\left(\frac{5}{6}\right)\right)}{16 \pi}$$
=
=
/ / -2*pi*I / pi*I\ 2*pi*I / 5*pi*I\\ \
| | ------- | ----| ------ | ------|| |
| | 2/3 / 2/3 pi*I\ 2/3 3 | 2/3 3 | 2/3 3 | 2/3 3 || |
| 3 ___ 2 | 2 *log\1 - 2 *e / 2 *e *log\1 - 2 *e / 2 *e *log\1 - 2 *e /| |
| 2*\/ 2 *Gamma (2/3)*|- ------------------------ - --------------------------------- - ----------------------------------|*Gamma(-1/6)*Gamma(1/3)*Gamma(7/6)|
2/3 | \ 6 6 6 / |
2 *|-pi*Gamma(-1/3)*Gamma(4/3)*log(5) + 2*pi*atan(2)*Gamma(1/6)*Gamma(5/6) + -----------------------------------------------------------------------------------------------------------------------------------------------------------|
\ Gamma(5/3) /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16*pi
$$\frac{2^{\frac{2}{3}} \left(\frac{2 \sqrt[3]{2} \left(- \frac{2^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} \log{\left(- 2^{\frac{2}{3}} e^{\frac{5 i \pi}{3}} + 1 \right)}}{6} - \frac{2^{\frac{2}{3}} \log{\left(1 - 2^{\frac{2}{3}} e^{i \pi} \right)}}{6} - \frac{2^{\frac{2}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(1 - 2^{\frac{2}{3}} e^{\frac{i \pi}{3}} \right)}}{6}\right) \Gamma\left(- \frac{1}{6}\right) \Gamma\left(\frac{1}{3}\right) \Gamma^{2}\left(\frac{2}{3}\right) \Gamma\left(\frac{7}{6}\right)}{\Gamma\left(\frac{5}{3}\right)} - \pi \log{\left(5 \right)} \Gamma\left(- \frac{1}{3}\right) \Gamma\left(\frac{4}{3}\right) + 2 \pi \operatorname{atan}{\left(2 \right)} \Gamma\left(\frac{1}{6}\right) \Gamma\left(\frac{5}{6}\right)\right)}{16 \pi}$$
2^(2/3)*(-pi*gamma(-1/3)*gamma(4/3)*log(5) + 2*pi*atan(2)*gamma(1/6)*gamma(5/6) + 2*2^(1/3)*gamma(2/3)^2*(-2^(2/3)*log(1 - 2^(2/3)*exp_polar(pi*i))/6 - 2^(2/3)*exp(-2*pi*i/3)*log(1 - 2^(2/3)*exp_polar(pi*i/3))/6 - 2^(2/3)*exp(2*pi*i/3)*log(1 - 2^(2/3)*exp_polar(5*pi*i/3))/6)*gamma(-1/6)*gamma(1/3)*gamma(7/6)/gamma(5/3))/(16*pi)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.