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