Problem $3\mathrm{.}1.$ Implement an adaptive $5-$point Gaussian quadrature rule as follows. Your function should take the following inputs

$1.$ a $<$ b endpoints of the integration interval $2.$ f function to integrate

$3.$ tol tolerance for integral accuracy

and return an approximation to the integral Rb f$($x$)$dx using the following procedure. a

$1.$ Initialize an $$in$-$progress$$ list with $[($a$1=$ a$,$ b$1=$ b$,$ t$1$ R b f $($x$)$dx$)],$ a numerical a

variable to keep track of the integral so far, and a $$done$$ list.

$2.$ While the $$in$-$progress$$ list is not empty, remove the last triple $($ak$,$bk$,$tk$)$ and use it

to produce two new triples $($ak$,$ ak$+$bk $,$ tl$)$ and $($ak$+$bk $,$ bk$,$ tr$)$ where integrals tl and tr are $22$

approximated using $5-$point Guassian quadrature.