Properties of the integrals (Formula presented.) of the Legendre polynomials P n(x) on the base interval -1 ≤ x ≤ 1 are systematically considered. The generating function (Formula presented.) is defined; here, Q 0 = 0 and Q k with k > 0 is a polynomial of degree 2k - 1 in each of the variables x and z. A second-order differential equation is derived, an analogue of Rodrigues' formula is obtained, and the asymptotic behavior as n → ∞ is determined. It is proved that the representation (Formula presented.) holds if and only if n ≥ k, where f nk is a polynomial divisible by neither x - 1 nor x + 1. The main result is the sharp bound (formula presented.) Here, (formula presented.) where t k is the maximum of the function √tJk(t) on the half-axis t > 0 and J k(t) is the Bessel function. The first values A k and differences A k - μ1 k 1/6 are tabulated below as follows.