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