I am aware that if $f(x)$ is a periodic function with period $T$ then:
1.) $\int^{nT}_{0}f(x)\,dx = n\int^{T}_{0}f(x)\,dx$, for $n$ an integer.
2.) $\int^{T+a}_{a}f(x)\,dx = \int^{T}_{0}f(x)\,dx$, for some constant $a$.
There are many wonderful and imaginative proofs of the above properties, but my question is this: Can you think of any particularly interesting integrals that would otherwise be very difficult to evaluate without these rules? I think it would be rather interesting to see how the properties can be used to simplify more complex problems. Thanks.
Well, showing that the integral $\int_0^ \infty |\sin x| / x dx$ diverges is not too hard to see from 2). You'd probably have a hard time proving this without it, but maybe this is more of a criticism of my lack of imagination than actual answer to your problem.