Calculate an integral $$ \int\limits_0^3 x^{-3/5}(3-x)^{-3/5} dx. $$ I know it may be done using gamma-function. But I can't find the appropriate substitute.
2026-04-02 21:38:06.1775165886
Calculate $\int\limits_0^3 x^{-3/5}(3-x)^{-3/5} dx$
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If you set $x=3u$, you get
$$3^{-1/5}\int_0^1u^{-3/5}(1-u)^{-3/5}\ du$$
Which is clearly a Beta integral, though the result does not simplify much.
$$\frac{[\Gamma(2/5)]^2}{\sqrt[5]{3}\Gamma(4/5)}$$