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