$($a$)$ What is Trapezoidal rule $?$ Using trapezoidal rule calculate the value of the integral ${\displaystyle \underset{4}{\overset{5\mathrm{.}2}{}}}lo{g}_{e}xdx,$ given that

$\backslash$table$[[x,4\mathrm{.}0,4\mathrm{.}2,4\mathrm{.}4,4\mathrm{.}6,4\mathrm{.}8,5\mathrm{.}0,5\mathrm{.}2],[lo{g}_{e}x,1\mathrm{.}3863,1\mathrm{.}4351,1\mathrm{.}4816,1\mathrm{.}5260,1\mathrm{.}5680,1\mathrm{.}6094,1\mathrm{.}6486]]$

Compare with exact value.

$($b$)$ Discuss extrapolation technique of Richardson and Gaunt.