Evaluate the following integral:

${\displaystyle \underset{1}{\overset{3}{}}}(2\mathrm{.}5+3\mathrm{.}8x^2-1\mathrm{.}6x^3+14\mathrm{.}2x^5-8\mathrm{.}1x^6)dx$

$($a$)$ analytically;

$($b$)$ single application of the trapezoidal rule;

$($c$)$ composite trapezoidal rule, with $n=2$ and $4$;

$($d$)$ single application of Simpson's $\frac{1}{3}$ rule;

$($e$)$ Simpson's $\frac{3}{8}$ rule;

$($f$)$ estimated the errors,

$2.$ Using the Backward, Forward, and, Centered method, estimated the derivative of

$f(x)=-0\mathrm{.}5x^5+0\mathrm{.}18x^4-1\mathrm{.}2x^3+0\mathrm{.}8x^2-2\mathrm{.}3$

at $x=0\mathrm{.}25$ using finite divided differences and a step size of $h=0\mathrm{.}05,$ Determine the errors.