Limit of a singular integral

Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider: $$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$ I would like to know the limit of $I(\gamma)$ when $\gamma \to \infty$, and if this limit is infinite I would like to have an equivalent.

My first idea was to say that $f_{\gamma}(x)$ converges to $\frac 1 2 (\delta_{-1}+\delta_1)$, and replacing this into the integral we obtain something which seems infinite. But I know the convergence is not uniform, so this argument is not valid.

Any idea?

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the question is in MO too –  what'sup Nov 2 '13 at 9:45