MathDB
Today's calculation of Integral 299

Source: 2008 Meiji University entrance exam/Science and Technology

February 16, 2008
calculusintegrationlogarithmsinductionalgebrapolynomialcalculus computations

Problem Statement

Let I_n(x)\equal{}\int_1^x (\ln t)^ndt\ (x>0) for n\equal{}1,\ 2,\ 3,\ \cdots. (1) Prove by mathematical induction that In(x) I_n(x) is expressed by I_n(x)\equal{}xf_n(\ln x)\plus{}C_n\ (n\geq 1) in terms of some polynomial fn(y) f_n(y) with degree n n and some constant number Cn. C_n. (2) Express the constant term of fn(y) f_n(y) interms of n. n.
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