In mathematics, an interval usually refers to a set of real numbers: if X and Y are two numbers in a set, then any number between X and Y also belongs to the set.

Interval plays an important role in integral theory, because as the simplest set of real numbers, they can be easily defined as "length" or "measure". Then we can extend the concept of "measure" and derive Borel measure and Lebesgue measure. Interval is also the core concept of interval arithmetic, which is a numerical analysis method used to calculate rounding error.

Introduction:

When trigonometric functions are mixed with complex exponential logarithms or ordinary polynomials (such as x* è Sinks è), and the integral region contains π/2, π, etc., the interval reproduction formula is suitable.

In this way, the integral region will not change, and the substitution of X in trigonometric function caused by variable substitution can be eliminated by inductive formula. The subtlety of interval regeneration formula is that the integrand function can be modified without changing the integral region.

This method of substitution is called the integral interval exchange formula (or the integral interval regeneration formula), and its essence is to replace the original integral variable X, even if x+t=a+b (a and B are the upper and lower limits of the original integral respectively), and replace X with T to become a new integral variable.