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While browsing MSE, I found some posts regarding integration tricks / integration formulae, for both definite and indefinite integrals.

I saw this post, this post, this post, and some other posts.

I saw the following (but not only the following),

I am familiar with many of them

Now I am asking about a book that includes such integration formulae with there proofs and examples.

$$\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$$

For rational expressions of trigonometric functions, substitute:

$$\sin(x)=\frac{2t}{1+t^2}, \tan(x)=\frac{2t}{1-t^2}, \sec(x)=\frac{1+t^2}{1-t^2}, \text{and } dx=\frac{2dt}{1+t^2}$$

This is called "tangent half-angle substitution", so these substitutions can be derived by first putting $\tan(x/2)=t$. This substitution also known as "Weierstrass substitution".

$$\int_{-a}^{a} f(x) dx = \left\{\begin{matrix} 2\int_{0}^{a} f(x) dx &, \text{when }f(x) \text{ is an even function} \\ \\ 0 &, \text{when }f(x) \text{ is an odd function} \\ \end{matrix}\right.$$

Integration of an inverse function:

$$\int f^{-1}(x)dx = x f^{-1}(x)-F(f^{-1}(x))+c, \text{where } F(x)=\int f(x)dx$$

Frullani Integral:

$$\int_{0}^{\infty} \frac{f(ax)-f(bx)}{x} dx = \bigg(f(\infty)-f(0)\bigg)\log\bigg(\frac{a}{b}\bigg)$$

$$\int_{-a}^{a} \frac{f_{1}(x)dx}{1 \pm \bigg(f_{2}(x)\bigg)^{f_{3}(x)}}=\int_{0}^{a}f_{1}(x)dx$$

provided that both $f_{1}$ and $f_{2}$ are even functions, and $f_{3}$ is an odd function.

Laplace Integration:

$$\int_{0}^{\infty} \frac{f(x)}{x}dx = \int_{0}^{\infty}\mathcal{L}\{f(t)\}ds$$

I need a book that includes (not only these) integrals. Hopefully (only one comprehensive) book.

Your help would be appreciated. THANKS!

Hussain-Alqatari
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    I’m afraid such an all encompassing book is unavailable.. – Quanto Jul 05 '22 at 14:01
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    @Quanto Hopefully, $\exists$ book $\in \mathcal{U}$niverse. – Hussain-Alqatari Jul 05 '22 at 14:05
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    As someone who has spent nearly 5 decades of browsing the math collections in dozens of large university libraries, among many other things, I think instead of looking for a comprehensive book on some topic or other (few of which actually exist, and those that do are often treatises intended for research reference purposes), you're better off assembling a collection of fairly comprehensive books on various portions of what you're interested in. That said, I assembled some books on integration techniques in this answer. – Dave L. Renfro Jul 05 '22 at 14:15

1 Answers1

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Such books are mostly from more than 50 years ago... Example

MacNeish, H. F., Algebraic technique of integration, (Publications in Mathematics. 1) Florida: University of Miami Press VIII, 109 p. (1950). ZBL0041.17803.

Here is a recent one, about 150 pages:

Markin, Marat V., Integration for calculus, analysis, and differential equations. Techniques, examples, and exercises, Hackensack, NJ: World Scientific (ISBN 978-981-3272-03-3/hbk; 978-981-3275-15-7/pbk). xii, 164 p. (2019). ZBL1395.00004.

Advanced methods would be Risch algorithm and similar. (Here the point is not doing it by hand, but programming a computer to do it.) Example:

Bronstein, Manuel, Symbolic integration. I: Transcendental functions, Algorithms and Computation in Mathematics. 1. Berlin: Springer. xiii, 299 pp. (1997). ZBL0880.12005.

GEdgar
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  • Thanks for the Markin reference, which I didn't know about. I've added it to my answer cited in a comment to the OP's question. – Dave L. Renfro Jul 05 '22 at 17:02