Sr Examen
Integral de sqrt((x^3)+2) dx
Solución
Ha introducido
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9 / | | ________ | / 3 | \/ x + 2 dx | / -1
$$\int\limits_{-1}^{9} \sqrt{x^{3} + 2}\, dx$$
Integral(sqrt(x^3 + 2), (x, -1, 9))
Respuesta (Indefinida)
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/ _ / | 3 pi*I\ | ___ |_ |-1/2, 1/3 | x *e | | ________ x*\/ 2 *Gamma(1/3)* | | | --------| | / 3 2 1 \ 4/3 | 2 / | \/ x + 2 dx = C + ---------------------------------------------- | 3*Gamma(4/3) /
$$\int \sqrt{x^{3} + 2}\, dx = C + \frac{\sqrt{2} x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{x^{3} e^{i \pi}}{2}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$
Gráfica
Respuesta
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_ / | pi*I\ _
___ |_ |-1/2, 1/3 | 729*e | ___ |_ /-1/2, 1/3 | \
3*\/ 2 *Gamma(1/3)* | | | ---------| \/ 2 *Gamma(1/3)* | | | 1/2|
2 1 \ 4/3 | 2 / 2 1 \ 4/3 | /
----------------------------------------------- + ---------------------------------------
Gamma(4/3) 3*Gamma(4/3)
$$\frac{\sqrt{2} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{1}{2}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{3 \sqrt{2} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{729 e^{i \pi}}{2}} \right)}}{\Gamma\left(\frac{4}{3}\right)}$$
=
=
_ / | pi*I\ _
___ |_ |-1/2, 1/3 | 729*e | ___ |_ /-1/2, 1/3 | \
3*\/ 2 *Gamma(1/3)* | | | ---------| \/ 2 *Gamma(1/3)* | | | 1/2|
2 1 \ 4/3 | 2 / 2 1 \ 4/3 | /
----------------------------------------------- + ---------------------------------------
Gamma(4/3) 3*Gamma(4/3)
$$\frac{\sqrt{2} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{1}{2}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{3 \sqrt{2} \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {\frac{729 e^{i \pi}}{2}} \right)}}{\Gamma\left(\frac{4}{3}\right)}$$
3*sqrt(2)*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), 729*exp_polar(pi*i)/2)/gamma(4/3) + sqrt(2)*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), 1/2)/(3*gamma(4/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.