The main motivation behind this paper is the question ‘What is the factorization of x ⁿ + y ⁿ when n is an even positive integer?’ which was and is frequently asked on the Internet by many high school and university students and, to my knowledge, even the specialized textbooks and research articles have not yet answered the question, and over time the question itself is transformed into a problem that needs to be solved. In the present article, the question is positively answered and the problem is solved through the detailed study of the factorization that leads directly to an apparently new type of indefinite irrational integrals.