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