Problem $1$

Evaluate the following integrals numerically:

Using Simpson's $\frac{1}{3}$ rule with $n=2$

Using Simpson's $\frac{1}{3}$ rule with $n=4$

Apply Richardson's extrapolation to find a new estimate of the integral using your results in parts $1$ and $2.$

$I={\displaystyle \underset{1}{\overset{2}{}}}(x+\frac{1}{x}{)}^{2}dx,I={\displaystyle \underset{0}{\overset{3}{}}}x{e}^{2x}dx$