In this paper, the Hilbert integral inequality \[ \int^\infty_0 \int^\infty_0 {f(x)g(y)\over x+y} dx dy\leq \pi\Biggl(\int^\infty_0 f^2(x)dx\Biggr)^{1/2}\Biggl(\int^\infty_0 g^2(x)dx\Biggr)^{1/2}\tag{1} \] and its equivalent form \[ \int^\infty_0 \Biggl(\int^\infty_0 {f(x)\over x+y} dx\Biggr)^2 dy\leq \pi^2\int^\infty_0 f^2(x)dx\tag{2} \] are generalized. Namely, the author investigates variants of (1), where its left-hand side is replaced by \(\int^b_a \int^b_a(f(x)g(y)/(x+ y)^t)dx dy\) (with \(0