Assuming you're talking about the [tex]n[/tex]th moment, [tex]E[X^n][/tex], we have by definition of expectation of a function of a random variable
[tex]\displaystyle E[X^n]=\int_{-\infty}^\infty x^nf_X(x)\,\mathrm dx[/tex]
where [tex]f_X[/tex] is the PDF for [tex]X[/tex]. The [tex]n[/tex]th moment for the standard uniform distribution is then
[tex]\displaystyle\int_0^1 x^n\,\mathrm dx=\frac{x^{n+1}}{n+1}\bigg|_0^1=\boxed{\frac1{n+1}}[/tex]
