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

$\doteq \frac{3\/10}{2}\left(f\left(1\right)\+f\left(4\right)\+2\sum _{k\=1}^{9}f\left(1\+\left(3\/10\right)k\right)\right)$

$\=5.067162904$

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 sum

$\mathrm{ApproximateInt}\left(F\,x\=1..4\,\mathrm{partition}\=10\,\mathrm{method}\=\mathrm{trapezoid}\,\mathrm{output}\=\mathrm{sum}\right)$

$\frac{3}{20}\sum _{i\=0}^9\left(2\+\sin \left(\sqrt{1\+\frac{3}{10}i}\ln \left(2\+\frac{3}{10}i\right)\right)\+\sin \left(\sqrt{\frac{13}{10}\+\frac{3}{10}i}\ln \left(\frac{23}{10}\+\frac{3}{10}i\right)\right)\right)$

Apply the ApproximateInt command with output set to value

$\mathrm{ApproximateInt}\left(F\,x\=1..4\,\mathrm{partition}\=10\,\mathrm{method}\=\mathrm{trapezoid}\,\mathrm{output}\=\mathrm{value}\right)$

$3\+\frac{3}{20}\sin \left(2\ln \left(5\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{10}\sqrt{31}\sqrt{2}\sqrt{5}\ln \left(\frac{41}{10}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{5}\sqrt{17}\sqrt{5}\ln \left(\frac{22}{5}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{10}\sqrt{37}\sqrt{2}\sqrt{5}\ln \left(\frac{47}{10}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{5}\sqrt{11}\sqrt{5}\ln \left(\frac{16}{5}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{2}\sqrt{5}\sqrt{2}\ln \left(\frac{7}{2}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{5}\sqrt{2}\sqrt{7}\sqrt{5}\ln \left(\frac{19}{5}\right)\right)\+\frac{3}{10}\sin \left(\frac{2}{5}\sqrt{2}\sqrt{5}\ln \left(\frac{13}{5}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{10}\sqrt{19}\sqrt{2}\sqrt{5}\ln \left(\frac{29}{10}\right)\right)\+\frac{3}{10}\sin \left(\frac{1}{10}\sqrt{13}\sqrt{2}\sqrt{5}\ln \left(\frac{23}{10}\right)\right)\+\frac{3}{20}\sin \left(\ln \left(2\right)\right)$

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

$\mathrm{ApproximateInt}\left(F\,x\=1..4.0\,\mathrm{partition}\=10\,\mathrm{method}\=\mathrm{trapezoid}\,\mathrm{output}\=\mathrm{value}\right)$ = $5.067162904$$$

Table 6.7.1(a)   Using the ApproximateInt command to implement the Trapezoid rule in Maple

$\mathrm{ApproximateInt}\left(f\left(x\right)\,x\=a..b\,\mathrm{partition}\=n\,\mathrm{method}\=\mathrm{trapezoid}\,\mathrm{output}\=\mathrm{sum}\right)$

$\frac{1}{2}\frac{\left(b-a\right)\left(\sum _{i\=0}^{n-1}\left(f\left(a\+\frac{i\left(b-a\right)}{n}\right)\+f\left(a\+\frac{\left(i\+1\right)\left(b-a\right)}{n}\right)\right)\right)}{n}$

$f\left(x\right)\=F$$\stackrel{\text{assign as function}}{\to }$$f$

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

$\frac{h}{2}\left(f\left(1.\right)\+f\left(4.\right)\+2\sum _{k\=1}^{9}f\left(1.\+k h\right)\right)$ = $5.067162905$$$

Table 6.7.1(b)   Implementing the Trapezoid rule from first principles