Let K= _ 1 ^ e x

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

K=\int_{1}^{e} x\,\ln x\,dx.

This integral arises when finding the centroid of the region bounded by

y=\ln x

[1,e]

. (a) Evaluate

K

exactly. (b) The average value of

\ln x

[1,e]

is given by

\dfrac{1}{e-1}\int_{1}^{e}\ln x\,dx

. Use the result of part (a) to compute this average value.