TL;DR: In this article, the authors present a method for the construction of integral identities that contain an undetermined function, except for mild restrictions, this function can be chosen arbitrarily, and their method is illustrated by several examples leading to new integral identities.

Abstract: We present a method for the construction of integral identities that contain an undetermined function. Except for mild restrictions, this function can be chosen arbitrarily. Our method is illustrated by several examples leading to new integral identities.

TL;DR: Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + bdx = 1 a ln|ax + b| (4) Integrals of Rational Functions

Abstract: Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + b dx = 1 a ln |ax + b| (4) Integrals of Rational Functions 1 (x + a) 2 dx = −

11,190 citations

"Construction of Generalized Integra..." refers background in this paper

...Existing resources for integral tables, such as the monographs (Abramowitz & Stegun, 1964; Brychkov, 2008; Gradshtevn & Ryzhik, 2015) or online data bases (Shapiro, 2015; Wolfram Alpha, 2016), are continuously extended as new methods for the resolution of integrals are discovered....

TL;DR: The Derivatives. Indefinite Integrals. Connection Formulas and the Meijer G Function as discussed by the authors are derived from Hypergeometric Functions and the connection formulas, respectively.

Abstract: The Derivatives. Limits. Indefinite Integrals. Definite Integrals. Infinite Series. The Connection Formulas. Representations of Hypergeometric Functions and the Meijer G Function.

293 citations

"Construction of Generalized Integra..." refers background in this paper

...Existing resources for integral tables, such as the monographs (Abramowitz & Stegun, 1964; Brychkov, 2008; Gradshtevn & Ryzhik, 2015) or online data bases (Shapiro, 2015; Wolfram Alpha, 2016), are continuously extended as new methods for the resolution of integrals are discovered....

"Construction of Generalized Integra..." refers background in this paper

...Existing resources for integral tables, such as the monographs (Abramowitz & Stegun, 1964; Brychkov, 2008; Gradshtevn & Ryzhik, 2015) or online data bases (Shapiro, 2015; Wolfram Alpha, 2016), are continuously extended as new methods for the resolution of integrals are discovered....

TL;DR: In this paper, it was shown that the integral of F(u) is the same as integral F(x), both with respect to x, for a variety of functions u.

Abstract: We show that the integral of F(u) is the same as the integral of F(x), both with respect to x, for a variety of functions u. In this note F will denote any integrable function, and all integral signs denote the Cauchy principal value. The formula /OO ,«00 F(u)dx= F(x)dx, -00 ■'-oo where (2) u = x — 1/x, goes back at least as far as Cauchy and can be proved as follows: From a graph of the relation (2) one sees that the solution for x has two branches 1 x± = -y \ u ± iju2 + 4 \, -oo

17 citations

"Construction of Generalized Integra..." refers methods in this paper

...An example for such a new method is presented in the recent work by Glasser (2013)....