Beta and Gamma Functions

Reflecting the interval of integration and applying Euler's reflection formula evaluates a definite integral involving a fifth root.

A homographic change of variables reduces a beta-type integral with a shifted linear denominator to Euler's beta function.

A change of variables and an inversion symmetry reduce integral 3.255 from Gradshteyn and Ryzhik to a beta integral.

A contour argument gives integral representations involving the reciprocal beta function and their logarithmic consequences.

Use Ramanujan's master theorem to connect power-series coefficients with Mellin transforms of Bessel functions.