Question$-2$: Estimate ${\displaystyle \underset{0}{\overset{2\mathrm{.}5}{}}}f(x)dx$ from the following data:

$\backslash$table$[[x,0,0\mathrm{.}5,1\mathrm{.}0,1\mathrm{.}5,2\mathrm{.}0,2\mathrm{.}5],[f(x),1\mathrm{.}5000,2\mathrm{.}0000,2\mathrm{.}0000,1\mathrm{.}6364,1\mathrm{.}2500,0\mathrm{.}9565]]$

a$)$ Use Simpson's $\frac{1}{3}$ Rule,

b$)$ Use Simpson's $\frac{3}{8}$ Rule,

c$)$ Use Simpson's $\frac{3}{8}$ Rule in conjunction with Simpson's $\frac{1}{3}$ Rule to integrate the same function for five segments.

d$)$ Calculate the true percent relative error for each case.