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We investigate the asymptotic distribution of integrals of the $j$-functionthat are associated to ideal classes in a real quadratic field. To estimate theerror term in our asymptotic formula, we prove a bound for sums of Kloostermansums of half-integral weight that is uniform in every parameter. To establishthis estimate we prove a variant of Kuznetsov's formula where the spectral datais restricted to half-integral weight forms in the Kohnen plus space, and weapply Young's hybrid subconvexity estimates for twisted modular $L$-functions.

Modular invariants for real quadratic fields and Kloosterman sums