Let $$T>0$$ be a fixed number. $$f: R \rightarrow R$$ is a continuous function such that $$f(x+T)=f(x), x \in R$$ If $$I=\int_\limits0^T f(x) d x$$, then $$\int_\limits0^{5 T} f(2 x) d x=$$

$$\int_\limits1^3 x^n \sqrt{x^2-1} d x=6 \text {, then } n=$$

[ . ] represents greatest integer function, then $$\int_{-1}^1(x[1+\sin \pi x]+1) d x=$$

$$\begin{aligned} & \lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2 n+1}}+\frac{n}{(n+2) \sqrt{2(2 n+2)}}\right. \\ & \left.+\frac{n}{(n+3) \sqrt{3(2 n+3)}}+\ldots n \text { terms }\right]=\int_\limits0^1 f(x) d x \end{aligned}$$

then $$f(x)=$$