In this article, using the trigonometric power formulae, the L’Hospital rule and integration by parts, some evaluations for the improper integrals  ∫∞0(sin x/x)^n dx and  ∫∞0(sin(βx)/x)^n dx for positive integer n ≥ 2 and β ≠ 0 are established. As by-products, three combinatorial identities are obtained.