Let f: R → R be a continuous function satisfying f(x)+f(x+ k )= n, for all x ∈ R where k >0 and n is a positive integer. If I_1=∫_0^4 n k f(x) d x…

Let $\displaystyle f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function satisfying $\displaystyle f(x)+f(x+\mathbf{k})=\mathbf{n}$, for all $\displaystyle x \in \mathbf{R}$ where $\displaystyle \mathrm{k}>0$ and n is a positive integer. If $\displaystyle \mathrm{I}_1=\int_0^{4 \mathrm{n} \mathrm{k}} f(x) \mathrm{d} x$ and $\displaystyle \mathrm{I}_2=\int_{-\mathrm{k}}^{3 \mathrm{k}} f(x) \mathrm{d} x$, then :