We study the exponential integrability problem for I-n(f(n)), i.e., E exp{I-n(f(n))} < infinity, where I-n(f(n)) is a multiple Ito-Wiener integral on some Gaussian space arising from the constructive quantum field theory and from stochastic quantization. We also study a class of singular infinite-dimensional stochastic differential equations whose drift coefficients are measurable and unbounded. Using a condition due to Kazamaki, we prove the existence of a weak solution assuming some integrability conditions on the drift coefficient. Then we apply the exponential integrability theorem and the existence theorem to study infinite-dimensional stochastic differential equations of stochastic quantization.