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Today's calculation of Integral 786

Source: 2012 Chuo University entrance exam/Science and Technology

February 15, 2012
calculusintegrationlimitinductionalgebrapolynomialprobability

Problem Statement

For each positive integer nn, define Hn(x)=(−1)nex2dndxne−x2.H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.
(1) Find H1(x), H2(x), H3(x)H_1(x),\ H_2(x),\ H_3(x).
(2) Express ddxHn(x)\frac{d}{dx}H_n(x) interms of Hn(x), Hn+1(x).H_n(x),\ H_{n+1}(x). Then prove that Hn(x)H_n(x) is a polynpmial with degree nn by induction.
(3) Let aa be real number. For n≥3n\geq 3, express Sn(a)=∫0axHn(x)e−x2dxS_n(a)=\int_0^a xH_n(x)e^{-x^2}dx in terms of Hn−1(a), Hn−2(a), Hn−2(0)H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0).
(4) Find lim⁡a→∞S6(a)\lim_{a\to\infty} S_6(a). If necessary, you may use lim⁡x→∞xke−x2=0\lim_{x\to\infty}x^ke^{-x^2}=0 for a positive integer kk.
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