Since the period of f(x) is t, so f(x)=f(x+T), let:

g(x)=∫_{x}^{x+T}f(t)dt

=∫_{0}^{x+t}f(t)dt-∫_{0}^{x}f(t)dt

So g' (x) = f (x+t)-f (x) = 0.

Therefore, g(x) is a constant function.

g(x)=g(0).

that is

∫_{x}^{x+t}f(t)dt=∫_{0}^{t}f(t)dt。

Definite integral is a kind of integral, which is the limit of the integral sum of function f(x) in the interval [a, b]. Here, we should pay attention to the relationship between definite integral and indefinite integral: if definite integral exists, it is a concrete numerical value, while indefinite integral is a functional expression, and they have only one mathematical relationship (Newton-Leibniz formula).

Brief introduction of definite integral

A function can have indefinite integral, but not definite integral; There can also be definite integral, but there is no indefinite integral. A continuous function must have definite integral and indefinite integral; If there are only a finite number of discontinuous points, the definite integral exists; If there is jump discontinuity, the original function must not exist, that is, the indefinite integral must not exist.