Derivation of Equilibrium Strategy in 1st-price Auction?
I think you have a typo -- the equilibrium bidding strategy in the first-price auction you specify should be $$ \beta(x) = \frac{N-1}{N}x $$

Here's a hint that might help. The CDF of the order statistic, $Y_1$, is $$G(x) = x^{N-1}$$

To see why this is the case, notice that this is exactly the probability that some given quantity $x$ is greater than or equal to $N-1$ independently drawn random variables that are uniformly distributed on $[0,1]$.

I'll write something in more detail when I have more time tomorrow, but I think this should tide you over for now. (At the very least, it'll help you evaluate the integral.)