Sr Examen
Integral de x/(1+x^3)^1/3 dx
Solución
Ha introducido
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0 / | | x | ----------- dx | ________ | 3 / 3 | \/ 1 + x | / 2
$$\int\limits_{2}^{0} \frac{x}{\sqrt[3]{x^{3} + 1}}\, dx$$
Integral(x/(1 + x^3)^(1/3), (x, 2, 0))
Respuesta (Indefinida)
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_ / 2 |_ /1/3, 2/3 | 3 pi*I\ | x *Gamma(2/3)* | | | x *e | | x 2 1 \ 5/3 | / | ----------- dx = C + ---------------------------------------- | ________ 3*Gamma(5/3) | 3 / 3 | \/ 1 + x | /
$$\int \frac{x}{\sqrt[3]{x^{3} + 1}}\, dx = C + \frac{x^{2} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {x^{3} e^{i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$$
Gráfica
Respuesta
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_
|_ /1/3, 2/3 | pi*I\
-4*Gamma(2/3)* | | | 8*e |
2 1 \ 5/3 | /
---------------------------------------
3*Gamma(5/3)
$$- \frac{4 \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {8 e^{i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$$
=
=
_
|_ /1/3, 2/3 | pi*I\
-4*Gamma(2/3)* | | | 8*e |
2 1 \ 5/3 | /
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3*Gamma(5/3)
$$- \frac{4 \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {8 e^{i \pi}} \right)}}{3 \Gamma\left(\frac{5}{3}\right)}$$
-4*gamma(2/3)*hyper((1/3, 2/3), (5/3,), 8*exp_polar(pi*i))/(3*gamma(5/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.