Let n≥2 be natural number. Answer the following questions.
(1) Evaluate the definite integral ∫1nxlnx dx.
(2) Prove the following inequality.
\frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.
(3) Find limn→∞(11⋅22⋅33⋯⋯nn)n2lnn1. calculusintegrationlogarithmslimitfunctioncalculus computations