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In connection with the non-uniqueness of the Stieltjes moment problem on (0,∞), Stieltjes constructed the nontrivial function f(x)=e^-x^1/4 \!(x^1/4) satisfying _0^∞ x^nf(x) dx=0 for all integers n0. We extend this by considering \[ I(k)=_0^∞ e^-x^1/4 \!(x^1/4) x^k dx \] for real k0, evaluating I(k) explicitly and proving I(k)=0 if and only if kZ_0. More generally, for parameters m>0, α>0, βR, q,kR we analyze \[ I_m,q(k;α,β)=_0^∞ e^-α x^1/m \!(β x^1/m) x^k(x^1/m)^q dx, \] derive a closed form, and give necessary and sufficient conditions for its vanishing. We also establish cosine analogues, both for the Stieltjes example and for the generalized integral mentioned above. As a consequence, we obtain integral representations of Γ(A) for suitable A > 0, as well as integral formulas for several classical constants arising from gamma function. To understand the importance of integrals that vanish for every value of a continuous parameter, we will also discuss Salem’s equivalence of the Riemann hypothesis, which is formulated in terms of such a parameter-dependent integral., number=4, journal=European Journal of Pure and Applied Mathematics, year=2025, month=Nov., pages=7023