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I've been trying to determine how certain definite integrals are expressed in terms of Gamma functions.
Mathematica returns the following:
[tex]\int_0^1 \frac{dx}{\sqrt{1-x^4}}=\frac{\sqrt{\pi}\Gamma[\frac{5}{4}]}{\Gamma[\frac{3}{4}]}[/tex]
(Mapple returns a different but equivalent expression in terms of Gamma)
In general it seems:
[tex]\int_0^1 \frac{dx}{\sqrt{1-x^n}}=\frac{\sqrt{\pi}\Gamma[\frac{n+1}{n}]}{\Gamma[\frac{n+2}{2n}]}[/tex]
Can anyone explain to me how this is determined or provide a hint or a reference?
Mathematica returns the following:
[tex]\int_0^1 \frac{dx}{\sqrt{1-x^4}}=\frac{\sqrt{\pi}\Gamma[\frac{5}{4}]}{\Gamma[\frac{3}{4}]}[/tex]
(Mapple returns a different but equivalent expression in terms of Gamma)
In general it seems:
[tex]\int_0^1 \frac{dx}{\sqrt{1-x^n}}=\frac{\sqrt{\pi}\Gamma[\frac{n+1}{n}]}{\Gamma[\frac{n+2}{2n}]}[/tex]
Can anyone explain to me how this is determined or provide a hint or a reference?
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