Using Simpsons, mid$-$ordinate and trapezoidal rules, kindly compute the results $to$ find out

which method $is$ most accurate and critically evaluated $of$ the numerical estimation

methods $by$ looking $at$ the accuracy $of$ the methods used.

${\displaystyle \underset{3\frac{}{2}}{\overset{5\frac{}{2}}{}}}2cosxdx$

Hint: $3\frac{}{2}=9\frac{}{6}$ and $5\frac{}{2}=15\frac{}{6},so$ the ordinates lie $at9\frac{}{6},10\frac{}{6},11\frac{}{6},$

$12\frac{}{6},13\frac{}{6},14\frac{}{6}$ and $15\frac{}{6}.$

The midpoint between $9\frac{}{6}$ and $10\frac{}{6}is(9\frac{}{6}+10\frac{}{6})2=19\frac{}{12}.$