Lobachevsky-type logarithmic integrals via partial-fraction series
Periodizing improper integrals via the partial-fraction expansion of cosecant squared reduces trigonometric-logarithmic integrals on the half-line to finite intervals.
A mathematics notebook
Tagged: trigonometric-integrals
Notes on integrals, infinite series, and number theory.
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A difference of arctangent integrals and dilogarithm identities
Poisson-kernel series expansions and two-variable dilogarithm relations evaluate a difference of arctangent integrals.
The inverse tangent integral at two minus root three
Integration by parts and the Fourier series of log tangent evaluate the inverse tangent integral at tan(pi/12) in terms of Catalan's constant.
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 trigonometric integral via a semicircular contour
A trigonometric substitution and integration around the right half of the unit disk reduce an inverse-sine integral to a beta function.
Mellin transforms of sine and cosine via a quarter-circle contour
Rotating the positive real axis to the positive imaginary axis via an indented quarter-circle contour evaluates the Mellin transforms of sine and cosine.