= Solution
Write $\lambda=1+\delta$, where $\delta\downarrow0$, and put $x=\lambda(1-u)$. The integration range is $0\leq u\leq1-1/\lambda=\delta+O(\delta^2)$, and
$$
\left((1-u)^{-2}-1\right)^{1/2}
=(2u)^{1/2}+O(u^{3/2}).
$$
Consequently
$$
\int_1^\lambda\left(\frac{\lambda^2}{x^2}-1\right)^{1/2}dx
=\frac{2\sqrt2}{3}\delta^{3/2}+O(\delta^{5/2}).
$$
Substitution into the quantization condition gives
$$
\boxed{\lambda=1+\alpha_n\epsilon^{2/3}+O(\epsilon^{4/3}),
\qquad
\alpha_n=\left[\frac{3\pi}{2\sqrt2}
\left(n+\frac34\right)\right]^{2/3}.}
$$
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