Sr Examen
Integral de x^n/sqrt(1-x^4) dx
Solución
Ha introducido
[src]
1 / | | n | x | ----------- dx | ________ | / 4 | \/ 1 - x | / 0
$$\int\limits_{0}^{1} \frac{x^{n}}{\sqrt{1 - x^{4}}}\, dx$$
Integral(x^n/sqrt(1 - x^4), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ 1 n | \
_ |1/2, - + - | |
n /1 n\ |_ | 4 4 | 4 2*pi*I|
/ x*x *Gamma|- + -|* | | | x *e |
| \4 4/ 2 1 | 5 n | |
| n | - + - | |
| x \ 4 4 | /
| ----------- dx = C + ------------------------------------------------
| ________ /5 n\
| / 4 4*Gamma|- + -|
| \/ 1 - x \4 4/
|
/
$$\int \frac{x^{n}}{\sqrt{1 - x^{4}}}\, dx = C + \frac{x x^{n} \Gamma\left(\frac{n}{4} + \frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{n}{4} + \frac{1}{4} \\ \frac{n}{4} + \frac{5}{4} \end{matrix}\middle| {x^{4} e^{2 i \pi}} \right)}}{4 \Gamma\left(\frac{n}{4} + \frac{5}{4}\right)}$$
Respuesta
[src]
/ 1 n | \
_ |1/2, - + - | |
/1 n\ |_ | 4 4 | |
Gamma|- + -|* | | | 1|
\4 4/ 2 1 | 5 n | |
| - + - | |
\ 4 4 | /
----------------------------------
/5 n\
4*Gamma|- + -|
\4 4/
$$\frac{\Gamma\left(\frac{n}{4} + \frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{n}{4} + \frac{1}{4} \\ \frac{n}{4} + \frac{5}{4} \end{matrix}\middle| {1} \right)}}{4 \Gamma\left(\frac{n}{4} + \frac{5}{4}\right)}$$
=
=
/ 1 n | \
_ |1/2, - + - | |
/1 n\ |_ | 4 4 | |
Gamma|- + -|* | | | 1|
\4 4/ 2 1 | 5 n | |
| - + - | |
\ 4 4 | /
----------------------------------
/5 n\
4*Gamma|- + -|
\4 4/
$$\frac{\Gamma\left(\frac{n}{4} + \frac{1}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{n}{4} + \frac{1}{4} \\ \frac{n}{4} + \frac{5}{4} \end{matrix}\middle| {1} \right)}}{4 \Gamma\left(\frac{n}{4} + \frac{5}{4}\right)}$$
gamma(1/4 + n/4)*hyper((1/2, 1/4 + n/4), (5/4 + n/4,), 1)/(4*gamma(5/4 + n/4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.