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!
