Expectation of the power of a uniform r.v.

Question

Assuming that $X$ is uniformly distributed on $[0,1]$, how do I derive that $$E \left( \frac{X^n}{n} \right) = \frac{1}{n(n+1)}?$$

Answer 1

Apply the definition of expected value: $$E \left( \frac{X^n}{n} \right) = \int_0^1 \left(\frac{x^n}{n} \right) \cdot 1 \, dx = \frac{1}{n(n+1)}$$