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