The following algorithm computes the integral $$\,J = \int\limits_a^b {f\left( x \right)dx\,\,\,} $$ from the given values $${f_j} = f\left( {{x_j}} \right)$$ at equidistant points $$\,\,{x_0} = a,\,\,{x_1} = {x_0} + h,\,\,$$ $$\,{x_2} = {x_0} + 2h,...\,{x_{2m}} = {x_0} + 2mh = b\,\,$$ compute
$${S_0} = {f_0} + {f_{2m}}$$
$${S_1} = {f_1} + {f_3} + .... + {f_{2m - 1}}$$
$${S_2} = {f_2} + {f_4} + .... + {f_{2m - 2}}$$

$$J = {h \over 3}\left[ {{S_0} + 4\left( {{S_1}} \right) + 2\left( {{S_2}} \right)} \right]$$

The rule of numerical integration, which uses the above algorithm is

Matching exercise choose the correct one out of the alternatives $$A, B, C, D$$

Group $$-$$ $${\rm I}$$
$$P.$$ $${2^{nd}}$$ order differential equations
$$Q.$$ Non-linear algebraic equations
$$R.$$ Linear algebraic equations
$$S.$$ Numerical integration

Group $$-$$ $${\rm II}$$
$$(1)$$ Runge $$-$$ Kutta method
$$(2)$$ Newton $$-$$ Raphson method
$$(3)$$ Gauss Elimination
$$(4)$$ Simpson's Rule

The real root of the equation $$x{e^x} = 2$$ is evaluated using Newton $$-$$ Raphson's method. If the first approximation of the value of $$x$$ is $$0.8679,$$ the $${2^{nd}}$$ approximation of the value of $$x$$ correct to three decimal places is