An integral involving the squared arctangent of sine squared
Integration by parts followed by differentiation under the integral sign reduces a squared-arctangent trigonometric integral to an elementary antiderivative.
A mathematics notebook
Tagged: definite-integrals
Notes on integrals, infinite series, and number theory.
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A real integral evaluated over a rectangular contour
Integrating a meromorphic function around a horizontal strip of height two pi evaluates a real integral mixing exponential and rational factors.
An algebraic integral with a fifth root via the beta function
Reflecting the interval of integration and applying Euler's reflection formula evaluates a definite integral involving a fifth root.
A parametric beta integral with a shifted linear denominator
A homographic change of variables reduces a beta-type integral with a shifted linear denominator to Euler's beta function.
A proof of GR 3.255
A change of variables and an inversion symmetry reduce integral 3.255 from Gradshteyn and Ryzhik to a beta integral.
Fourier Series of the Log-Gamma Function and Vardi's Integral
Derive Kummer's Fourier series for the logarithm of the gamma function, then use it to evaluate Vardi's integral.
Bernoulli numbers and a related integral
From the defining recursion for Bernoulli polynomials to their Fourier expansions and an associated integral.