Consider the following pseudo$-$code.

For $i$ in $1$ to $5000$ :

I. Generate $U_{i}Unif[0,1].$

II$.$ Set $x_i=-log(U_i).$

III. Evaluate $h(x_i)=\frac{x_i}{(1+x_i^2{)}^2{e}^{-x_i}}.$

Compute hat$()=\frac{1}{5000}{\displaystyle \underset{i=1}{\overset{5000}{}}}h(x_i).$

$($a$)(4$ points$)$ What is the integral that hat$()$ is estimating? Your answer should be of the form

${\displaystyle \underset{a}{\overset{b}{}}}g(x)dx$

where $a,binR\{-,\}.$

$($b$)(4$ points$)$ Implement the pseudo$-$code. Can you find any outliers?

$($c$)(6$ points$)$ Revise the pseudo$-$code for a better result with the same computation efficiency. Please provide your idea and pseudo$-$code here, and compare your results with part $($b$).$