Question 4 - A-Level Maths

Explain why \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) is an improper integral.
(1 mark)
Use integration by parts to find \(\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\).
(3 marks)
Show that \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) exists and find its value.
(4 marks)