Today's calculation of Integral 295
Source: 2008 Doshisya University entrance exam
February 10, 2008
calculusintegrationalgebrapolynomialgeometryfunctioncalculus computations
Problem Statement
Let f_n(x) \equal{} x^n(1 \minus{} x)^n,\ I_n \equal{} \int_0^1 f_n(x)\ dx\ (n \equal{} 1,2,\cdots).
(1) Find a polynomial such that f_{n \plus{} 1}'(x) \equal{} (n \plus{} 1)f_n(x)g(x).
(2) Find constant numbers such that f_{n \plus{} 2}''(x) \equal{} A_nf_{n \plus{} 1}(x) \plus{} B_nf_n(x).
(3) Find A_nI_{n \plus{} 1} \plus{} B_nI_n.
(4) Let J_n \equal{} (2n \plus{} 1)!I_n. Express J_{n \plus{} 1} in terms of .
(5) Find .