Sources for reading about integration techniques

Question

I wanted to know about techniques like Contour Integration, Leibniz Rule and so on. Complex analysis textbooks, like other analysis textbooks, only discuss generalisations and simple integration using learned material. I came across the Leibniz rule somehow, but a systematic treatment in some textbook could not be found. So, I wanted to know if textbooks or papers exist which explain integration through these and other advanced techniques, as well as discuss integrals with fair amount of complexity that can be solved.

Answer 1

Below are some books I know about, most of which I have a copy of (exceptions are Boros, Nahin, Vălean) and have found useful. Possibly more things of interest can be found in my answers to How to integrate $ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx$? (besides the list of references at the end, there are many other items cited throughout that are not specifically in the reference list). Also of possible interest is my answer to Computing the primitive $\;\int\frac{1}{1+x^n} dx$ and my answer to Integrate $\;(x^2+1)^\frac{1}{3}$.

Boros, Irresistible Integrals (2004)

Bromwich, Elementary Integrals (1911)

Edwards, A Treatise on the Integral Calculus (Volume I, 1921)

Ellingson, Integrals Containing a Parameter (1927 MS thesis; see comments here)

Gunther, Integration by Trigonometric and Imaginary Substitution (1907)

Fichtenholz, The Indefinite Integral (1971)

Hardy, The Integration of Functions of a Single Variable (2nd edition, 1916)

MacNeish, Algebraic Technique of Integration (1952)

Markin, Integration for Calculus, Analysis, and Differential Equations (2018)

Nahin, Inside Interesting Integrals (2nd edition, 2020)

Stewart, How to Integrate It (2018)

Vălean, (Almost) Impossible Integrals, Sums, and Series (2019)

Answer 2

Richard Courant's Introduction to calculus and analysis is a good book .It describes all fundamental formulas and then discusses all elementary techniques along with basic improper integrals .It introduces the Foundational Concepts of function and limit. Comprehensive coverage of the basics of integrals and differentials includes their applications as well as clearly defined techniques and essential theorems.