Integral de dx/sqrt(1-x^4) dx
Solución
Respuesta (Indefinida)
[src]
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/ |_ /1/4, 1/2 | 4 2*pi*I\
| x*Gamma(1/4)* | | | x *e |
| 1 2 1 \ 5/4 | /
| ----------- dx = C + -----------------------------------------
| ________ 4*Gamma(5/4)
| / 4
| \/ 1 - x
|
/
$$\int \frac{1}{\sqrt{1 - x^{4}}}\, dx = C + \frac{x \Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle| {x^{4} e^{2 i \pi}} \right)}}{4 \Gamma\left(\frac{5}{4}\right)}$$
_ _
|_ /1/4, 1/2 | \ |_ /1/4, 1/2 | \
Gamma(1/4)* | | | 1/16| Gamma(1/4)* | | | 1|
2 1 \ 5/4 | / 2 1 \ 5/4 | /
- --------------------------------- + ------------------------------
8*Gamma(5/4) 4*Gamma(5/4)
$$- \frac{\Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle| {\frac{1}{16}} \right)}}{8 \Gamma\left(\frac{5}{4}\right)} + \frac{\Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle| {1} \right)}}{4 \Gamma\left(\frac{5}{4}\right)}$$
=
_ _
|_ /1/4, 1/2 | \ |_ /1/4, 1/2 | \
Gamma(1/4)* | | | 1/16| Gamma(1/4)* | | | 1|
2 1 \ 5/4 | / 2 1 \ 5/4 | /
- --------------------------------- + ------------------------------
8*Gamma(5/4) 4*Gamma(5/4)
$$- \frac{\Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle| {\frac{1}{16}} \right)}}{8 \Gamma\left(\frac{5}{4}\right)} + \frac{\Gamma\left(\frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle| {1} \right)}}{4 \Gamma\left(\frac{5}{4}\right)}$$
-gamma(1/4)*hyper((1/4, 1/2), (5/4,), 1/16)/(8*gamma(5/4)) + gamma(1/4)*hyper((1/4, 1/2), (5/4,), 1)/(4*gamma(5/4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.