${\int }_1^4\left(1\+\sin \left(\sqrt{x}\ln \left(x\+1\right)\right)\right)$

$\doteq \frac{3\/10}{3}\left(f\left(1\right)\+f\left(4\right)\+2\sum _{k\=1}^{4}f\left(1\+\frac{3}{10}\left(2 k\right)\right)\+4\sum _{k\=1}^{5}f\left(1\+\frac{3}{10}\left(2 k-1\right)\right)\right)$

$\=5.078118675$

Evaluation via Maple's built-in numeric integrator

${\int }_1^4\left(1\+\sin \left(\sqrt{x}\ln \left(x\+1\right)\right)\right) \mathit{\ⅆ}x$$\stackrel{\text{at 10 digits}}{\to }$$5.078061188$ 

Initialize

Loading Student:-Calculus1

Context Panel: Assign to a Name≻$F$ 

$1\+\sin \left(\sqrt{x}\ln \left(x\+1\right)\right)$$\stackrel{\text{assign to a name}}{\to }$$F$

Apply the ApproximateInt command with output set to value, and the right endpoint a float

$\mathrm{ApproximateInt}\left(F\,x\=1..4.0\,\mathrm{partition}\=10\,\mathrm{method}\=\mathrm{simpson}\,\mathrm{partitiontype}\=\mathrm{normal}\right)$ = $5.078118675$$$

Table 6.7.4(a)   Using the ApproximateInt command to implement Simpson's rule in Maple

$f\left(x\right)\=1\+\sin \left(\sqrt{x}\ln \left(x\+1\right)\right)$$\stackrel{\text{assign as function}}{\to }$$f$

$h\=\frac{4-1}{10}$$\stackrel{\text{assign}}{\to }$

$\frac{h}{3}\left(f\left(1.\right)\+f\left(4.\right)\+2\sum _{k\=1}^{4}f\left(1.\+2 k h\right)\+4\sum _{k\=1}^{5}f\left(1.\+\left(2 k-1\right) h\right)\right)$ = $5.078118673$$$

Table 6.7.4(b)   Implementing Simpson's rule from first principles