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Today's calculation of Integral 661

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January 1, 2011
calculusintegrationlogarithmscalculus computations

Problem Statement

Consider a sequence 10.01, 20.02, 20.02, 30.03, 30.03, 30.03, 40.04, 40.04, 40.04, 40.04, ⋯1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots.
(1) Find the 36th term.
(2) Find ∫x2ln⁡x dx\int x^2\ln x\ dx.
(3) Let AA be the product of from the first term to the 36th term. How many digits does AA have integer part?
If necessary, you may use the fact 2.0<ln⁡8<2.1, 2.1<ln⁡9<2.2, 2.30<ln⁡10<2.312.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31.
2010 National Defense Medical College Entrance Exam, Problem 4
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