Sr Examen
Integral de x/(sqrt(x^5+1)) dx
Solución
Ha introducido
[src]
oo / | | x | ----------- dx | 2 | / 5 \ | t*\x + 1/ | / 2
$$\int\limits_{2}^{\infty} \frac{x}{t \left(x^{5} + 1\right)^{2}}\, dx$$
Integral(x/((t*(x^5 + 1)^2)), (x, 2, oo))
Respuesta
[src]
/ / pi*I\ / 7*pi*I\ / 3*pi*I\ / 9*pi*I\\
| -3*pi*I | ----| -pi*I | ------| pi*I | ------| 3*pi*I | ------||
| / pi*I\ ------- | 5 | ------ | 5 | ---- | 5 | ------ | 5 ||
| | e | 5 | e | 5 | e | 5 | e | 5 | e ||
| 6*log|1 - -----| 6*e *log|1 - -----| 6*e *log|1 - -------| 6*e *log|1 - -------| 6*e *log|1 - -------||
| 8 \ 2 / \ 2 / \ 2 / \ 2 / \ 2 /|
4*|- --- + ---------------- - ------------------------- - -------------------------- - ------------------------ - --------------------------|*Gamma(8/5)
\ 165 25 25 25 25 25 /
--------------------------------------------------------------------------------------------------------------------------------------------------------
5*t*Gamma(13/5)
$$\frac{4 \left(- \frac{6 e^{- \frac{i \pi}{5}} \log{\left(1 - \frac{e^{\frac{7 i \pi}{5}}}{2} \right)}}{25} - \frac{6 e^{\frac{3 i \pi}{5}} \log{\left(1 - \frac{e^{\frac{9 i \pi}{5}}}{2} \right)}}{25} - \frac{8}{165} + \frac{6 \log{\left(1 - \frac{e^{i \pi}}{2} \right)}}{25} - \frac{6 e^{- \frac{3 i \pi}{5}} \log{\left(1 - \frac{e^{\frac{i \pi}{5}}}{2} \right)}}{25} - \frac{6 e^{\frac{i \pi}{5}} \log{\left(1 - \frac{e^{\frac{3 i \pi}{5}}}{2} \right)}}{25}\right) \Gamma\left(\frac{8}{5}\right)}{5 t \Gamma\left(\frac{13}{5}\right)}$$
=
=
/ / pi*I\ / 7*pi*I\ / 3*pi*I\ / 9*pi*I\\
| -3*pi*I | ----| -pi*I | ------| pi*I | ------| 3*pi*I | ------||
| / pi*I\ ------- | 5 | ------ | 5 | ---- | 5 | ------ | 5 ||
| | e | 5 | e | 5 | e | 5 | e | 5 | e ||
| 6*log|1 - -----| 6*e *log|1 - -----| 6*e *log|1 - -------| 6*e *log|1 - -------| 6*e *log|1 - -------||
| 8 \ 2 / \ 2 / \ 2 / \ 2 / \ 2 /|
4*|- --- + ---------------- - ------------------------- - -------------------------- - ------------------------ - --------------------------|*Gamma(8/5)
\ 165 25 25 25 25 25 /
--------------------------------------------------------------------------------------------------------------------------------------------------------
5*t*Gamma(13/5)
$$\frac{4 \left(- \frac{6 e^{- \frac{i \pi}{5}} \log{\left(1 - \frac{e^{\frac{7 i \pi}{5}}}{2} \right)}}{25} - \frac{6 e^{\frac{3 i \pi}{5}} \log{\left(1 - \frac{e^{\frac{9 i \pi}{5}}}{2} \right)}}{25} - \frac{8}{165} + \frac{6 \log{\left(1 - \frac{e^{i \pi}}{2} \right)}}{25} - \frac{6 e^{- \frac{3 i \pi}{5}} \log{\left(1 - \frac{e^{\frac{i \pi}{5}}}{2} \right)}}{25} - \frac{6 e^{\frac{i \pi}{5}} \log{\left(1 - \frac{e^{\frac{3 i \pi}{5}}}{2} \right)}}{25}\right) \Gamma\left(\frac{8}{5}\right)}{5 t \Gamma\left(\frac{13}{5}\right)}$$
4*(-8/165 + 6*log(1 - exp_polar(pi*i)/2)/25 - 6*exp(-3*pi*i/5)*log(1 - exp_polar(pi*i/5)/2)/25 - 6*exp(-pi*i/5)*log(1 - exp_polar(7*pi*i/5)/2)/25 - 6*exp(pi*i/5)*log(1 - exp_polar(3*pi*i/5)/2)/25 - 6*exp(3*pi*i/5)*log(1 - exp_polar(9*pi*i/5)/2)/25)*gamma(8/5)/(5*t*gamma(13/5))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.