The integrals in Gradshteyn and Rhyzik. Part 1: A family of logarithmic integrals

The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of Logarithms and Exponentials

We present the evaluation of a family of exponential-logarithmic integrals. These have integrands of the form P(exp(x),ln(x)) where P is a polynomial. The examples presented here appear in sections 4.33, 4.34 and 4.35 in the classical table of integrals by I. Gradshteyn and I. Ryzhik.

Gradshteyn 与 Ryzhik 表中的双曲函数定积分推导

Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function

本文利用围道积分和特殊函数的解析延拓，推导 Gradshteyn 与 Ryzhik 表中的双曲函数定积分及相关无穷和，并以对数函数乘三角函数的积分为例，得到 Catalan 常数和 π 等结果。

Simple evaluation of one of Malmstén's integrals

The logarithmic integral no. 4.325.7 from Gradshteyn and Ryzhik's tables of integrals was first evaluated by Malmstén. Recently, Blagouchine used contour integration methods to evaluate a family of logarithmic integrals that contains this integral. We evaluate the integral in a simple, straightforward manner mainly by means of real analysis. The main ingredients of the evaluation are the use of geometric series and Kummer's Fourier series expansion for the logarithm of the gamma function.