In this paper, the authors establish some integral inequalities in two independent variables. These inequalities complement the recent results obtained by B. G. Pachpatte [JIPAM, J. Inequal. Pure Appl. Math. 2, No. 2, Paper 15 (2001; Zbl 0989.26011)]. Specifically, four major results are obtained in this paper but we choose to state one of the results to convey the importance of the paper: Let be nonnegative continuous functions defined for and satisfies (i) is positive, nondecreasing and continuous for , (ii) , , If the function is defined by
with nondecreasing in and for and
for and , then
is the inverse function of and
Other inequalities obtained in this paper are similar to the above result and the methods of proof are also similar. Applications of these results in obtaining boundedness and uniqueness of solutions to some partial differential equations are also given.