  In this paper we use the regularized Siegel-Weil formula of Gan-Qiu-Takeda to
obtain an expression of the integral of the theta function over the truncated
modular curve. We apply this result to express the integral over the truncated
modular curve of the logarithm of the Borcherds form and we describe explicitly
its asymptotic behaviour, and in particular the convergent and divergent
contributions. The result provides a complement to the work of Kudla on
integrals of Borcherds forms in a limiting case which falls out the range of
applications.
