Integrals, series,
and the ideas between.

Periodizing improper integrals via the partial-fraction expansion of cosecant squared reduces trigonometric-logarithmic integrals on the half-line to finite intervals.

Poisson-kernel series expansions and two-variable dilogarithm relations evaluate a difference of arctangent integrals.

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.

Integration by parts followed by differentiation under the integral sign reduces a squared-arctangent trigonometric integral to an elementary antiderivative.

Differentiating the Mellin transform of the sine function with respect to its exponent evaluates a logarithmic Fresnel-type integral in terms of Euler's constant.

Integrating a complex power around a keyhole cut along the negative real axis evaluates a rational integral involving the sine of a logarithm.

A trigonometric substitution and integration around the right half of the unit disk reduce an inverse-sine integral to a beta function.

Rotating the positive real axis to the positive imaginary axis via an indented quarter-circle contour evaluates the Mellin transforms of sine and cosine.

Residue calculus on a rectangular contour shows that the hyperbolic secant is its own Fourier transform up to scaling.

Integrating a meromorphic function around a horizontal strip of height two pi evaluates a real integral mixing exponential and rational factors.