Logarithmic power integrals

5 It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x$$

1. Show that, for \(n \geqslant 1\), $$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
2. Find \(I _ { 3 }\) in terms of e.

4 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x$$ for \(n \geqslant 0\). Show that, for all \(n \geqslant 1\), $$I _ { n } = \frac { 1 } { 3 } \mathrm { e } ^ { 3 } - \frac { 1 } { 3 } n I _ { n - 1 }$$ Find the exact value of \(I _ { 3 }\).

9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\).

1. Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2 .$$
2. Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\). [6]

9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\). Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2$$ Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\).