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We extend Borcherds' singular theta lift in signature $(1,2)$ to harmonicMaass forms of weight $1/2$ whose non-holomorphic part is allowed to be ofexponential growth at $i\infty$. We determine the singularities of the lift andcompute its Fourier expansion. It turns out that the lift is continuous but notdifferentiable along certain geodesics in the upper half-plane corresponding tothe non-holomorphic principal part of the input. As an application, we obtain ageneralization to higher level of the weight $2$ modular integral of Duke,Imamoglu and T\'oth. Further, we construct automorphic products associated toharmonic Maass forms.

Borcherds lifts of harmonic Maass forms and modular integrals