voko said:Recursive means something defined in terms of itself, but with a different conditions. For example, factorial: n! = (n - 1)!n.
It is not entirely clear what in you book is called "reduction formulas", but if that means, for example, something with degrees defined in terms of something similar with smaller degrees, then this could be equally called recursive formulas.
Recursion in integration refers to the process of breaking down an integral into smaller, simpler integrals and then solving them separately. This method involves repeatedly applying a specific formula or technique until the entire integral is solved.
Reduction in integration involves transforming a given integral into a simpler form that can be more easily solved. This is done by using various techniques such as substitution, integration by parts, or trigonometric identities. Reduction is often used when an integral cannot be solved by traditional methods.
While both recursion and reduction involve breaking down an integral into smaller parts, they differ in their approach. Recursion involves breaking down the integral into smaller integrals that are then solved separately, while reduction involves transforming the integral into a simpler form that can be solved using known techniques.
The efficiency of a method for solving integrals depends on the type of integral being solved. In some cases, recursion may be more efficient as it involves simpler calculations, while in other cases, reduction may be more efficient as it allows for the use of known techniques. Both methods have their advantages and disadvantages, and the most efficient approach will vary depending on the specific integral being solved.
Yes, recursion and reduction can be used together in integration. For example, a complex integral may first be simplified using reduction techniques and then solved using recursion by breaking it down into smaller integrals. This combination of methods can often be more effective in solving difficult integrals.