$($a$)$ Let $a2$ be a real number. By considering appropriate trapeziums, show that

${\displaystyle \underset{a-\frac{1}{2}}{\overset{a+\frac{1}{2}}{}}}lnxdxlna,$ and $,{\displaystyle \underset{a-1}{\overset{a}{}}}lnxdx\frac{ln(a-1)+lna}{2}$

$($b$)$ Using the result from $($a$),$ deduce that

${\displaystyle \underset{\frac{3}{2}}{\overset{n}{}}}lnxdxln(n!)-\frac{1}{2}lnn{\displaystyle \underset{1}{\overset{n}{}}}lnxdx$

for every integer $n2.$