# Periodic function and substitution question!

Let f: R-> R be a continuous function. Let T>0 be such that
f(x+T)= f(x) for all x.
We say that f is a periodic function with a period T>0.
Use an appropriate substitution to prove that for all real numbers a
$$\int^{a+T}_{a}f(x)dx$$ = $$\int^{T}_{0}f(x)dx$$.

I have no idea how to do this question.
thanks for helping me!

## Answers and Replies

jbunniii
Science Advisor
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Let f: R-> R be a continuous function. Let T>0 be such that
f(x+T)= f(x) for all x.
We say that f is a periodic function with a period T>0.
Use an appropriate substitution to prove that for all real numbers a
$$\int^{a+T}_{a}f(x)dx$$ = $$\int^{T}_{0}f(x)dx$$.

I have no idea how to do this question.
thanks for helping me!

Try breaking the integral on the left into two parts:

$$\int_{a}^{a+T} = \int_{a}^{T} + \int_{T}^{T+a}$$

and in the second part, use the fact that f is periodic.