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I am looking for an extensive source (often called "table of integrals") listing primitives of various classes of functions including the "elementary" ones (rational functions, functions involving radicals, ...), indicating also a/the convenient method to obtain the result.

For example, one entry could be similar to

$$\int x^2e^x\mathrm{d}\!x=e^x(x^2-2x+2)+\mathrm{Constant}$$

with the indication:

$$\textrm{By parts}\; (x^2,e^x); \textrm{by parts}\;(x,e^x)$$

or "integrate by parts two times" or just "by parts".

If there is a substitution to perform, it should be explicitely indicated, etc.

A free and user friendly computer program would also do. But for example Wolfram Alpha doesn't seem to "do the steps" or to give indications on the procedure.

Edit. Please note that

1) the example integral that I wrote above is, indeed, merely an example, not an indication of the intended complexity (or not) of the integrals!

2) I am not looking for textbooks that explain integration techniques (though I realize the distinction may not be sharp). I am merely looking for a list of primitives, endowed with indications about how each result has been obtained. Hypothetically, to explain how to obtain the primitive in my (easy) example above just literally one or two words may be enough.

3) I do not compute integrals all day long. I may very well forget which is the right substitution to perform in a certain instance. I am just looking for an easy way to get: "this is the result ok, and -aha- this is the method. So if I wanted to do the integral by myself I would know, without trying a bunch of useless substitutions/methods".

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I don't think there exists a published reference book that goes into as much detail as you want with the kind of very elementary integral you gave as an example (a simple exponential times a polynomial, which is the protype for the tabular method), since the example you gave would be well within the assumed background for a reader and would be too time consuming and too cluttering to be worth repeatedly mentioning (in a publication). Indeed, for the google search in my last sentence, I got nearly 9 million hits for the integration of parts tabular method. However, what you’re asking for DOES seem reasonable for software, even at the present time, and if something like this doesn't yet exist, I bet that it will within 10 years.

Anyway, a once well-known book, Peirce [7] (FYI, I have a hardback copy of the 1957 edition, revised by Ronald Martin Foster) gives brief descriptions of many integration methods. The descriptions appear at the beginning of many of the various chapters and sections. In fact, the reason I got a copy of this book (see here if you’re interested in getting a copy) was because I kept seeing it cited over and over again in older literature.

There are literally hundreds of freely available calculus books from the 1800s and early 1900s that provide extensive details on various integration methods, especially methods that have been edited out of calculus texts in the last 30 to 40 years. A few of the more comprehensive treatments I know about are [1], [2], [5], [8], [9], [10] below.

As for books that are primarily devoted to methods of finite form integration, the best such books I know about (that are not designed for researchers in present day computer methods of finite form integration) are Fichtenholz [3] (the only item below that, as far as I can tell, is not freely available on the internet), Hardy [4], and MacNeish [6].

Regarding the items below, the dates and links are to the latest edition I could find freely available on the internet. I’m using the latest editions to minimize typos and errors and such that an early edition might have. In some cases (especially Todhunter’s book), the date is much later than when the book first appeared.

[1] Joseph Edwards, A Treatise on the Integral Calculus, Volume 1 (1921)

[2] Joseph Edwards, A Treatise on the Integral Calculus, Volume 2 (1922)

[3] Grigorii Mikhailovich Fichtenholz, The Indefinite Integral (1971)

[4] Godfrey Harold Hardy, The Integration of Functions of a Single Variable (1916)

[5] Horace Lamb, An Elementary Course of Infinitesimal Calculus (1942)

[6] Harris Franklin MacNeish, Algebraic Technique of Integration (1952)

[7] Benjamin Osgood Peirce, A Short Table of Integrals (1914)

[8] William Benjamin Smith, Infinitesimal Analysis (1898)

[9] Isaac Todhunter A Treatise on the Integral Calculus (1906)

[10] Benjamin Williamson, An Elementary Treatise on the Integral Calculus (1888)

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