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!
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.