In this paper we have evaluated the integrals ∫ 0 ∞ x n − 1 ln x exp ( − a x − b x − 1 ) d x and ∫ 0 ∞ x n − 2 ( a x 2 − b ) ( ln x ) 2 exp ( − a x − b x − 1 ) d x for all n = 1 , 2 , 3 , … . Some applications of the results are discussed and an open problem is posed.