Question: Let a $f(x)$ be a periodic function. Another periodic function $g(x)$ can be obtained by compressing $f(x)$ by $1/k_1$ times in each of the period along the x axis and then magnifying it by $k_2$ times along the y axis.
If $f(x)=e^{\{x\}^2}(1+2{\{x\}}^2)$, where {} stands for factional part of x function. And $k_1,k_2$ satisfies $k_1^2-4k_1+k_2^2-6k_2+13=0$. Find the value of $$\int_0^{500}g(x)dx$$.
Firstly I figured out by graphs that our required function $g(x)=k_2\cdot f(k_1x)$. And easily solving the $k_1,k_2$ equation by forming squares we get $k_1=2, k_2=3$.
Now so we need to solve this difficult problem:
$$3\int_0^{500}e^{\{2x\}^2}(1+2{\{2x\}}^2)dx$$
Now I can't think of even how to begin solving. Breaking this into piecewise functions become extremely tedious and not worth solving. So I need help here. All suggestions are welcomed.
Hint: since $g(x)$ has period $1/2$, then $$ \int_0^{1/2}g(x)dx = \int_{n/2}^{(n+1)/2}g(x). $$ Can you split the integral from $0$ to $500$ into integrals of this form?