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Today's calculation of Integral 308

Source: 2008 Osaka Prefecture University entrance exam/Aeronautics

March 11, 2008
calculusintegrationfunctionlimitinductioncalculus computations

Problem Statement

Let a a be a positive constant number. For a positive integer n n, define a function In(t) I_n(t) by I_n(t)\equal{}\int_0^t x^ne^{\minus{}ax}dx. Answer the following questions. Note that you may use \lim_{t\rightarrow \infty} t^ne^{\minus{}at}\equal{}0 without proof. (1) Evaluate I1(t) I_1(t). (2) Find the relation of I_{n\plus{}1}(t),\ I_n(t). (3) Prove that there exists lim⁡t→∞In(t) \lim_{t\to\infty} I_n(t) for all natural number n n by using mathematical induction. (4) Find lim⁡t→∞In(t) \lim_{t\to\infty} I_n(t).
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