On a closed n-dimensional Riemannian manifold, assuming that the L^1 Kato integral of the Ricci curvature tensor is bounded, we prove a Souplet-Zhang type gradient estimate for bounded positive solutions of the heat equation. Then, by implanting the Souplet-Zhang type estimate in an argument of Qi S. Zhang, we show that certain integral Li-Yau inequality holds for the heat equation in this circumstance. The curvature assumption includes the case where the Ricci curvature tensor is L^p (p>n) integrable and volume is noncollapsing. This is a joint work with Xingyu Song, Ling Wu, and Qi S. Zhang