Explain why \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) is an improper integral.
Show that the substitution \(y = \frac { 1 } { x }\) transforms \(\int \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) into \(\int 2 y \ln y \mathrm {~d} y\).
Evaluate \(\int _ { 0 } ^ { 1 } 2 y \ln y \mathrm {~d} y\), showing the limiting process used.
Hence write down the value of \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\).