Question

By Simpson's rule, the value of ${\int }_1^2\frac{dx}{x}$ dividing the interval $(1,2)$ into four equal parts, is

• A. $0.6932$
• B. $0.6753$
• C. $0.6692$
• D. $7.1324$

Answer 1

Answer: (A)

$0.6932$

$h=\frac{2-1}{4}=\frac{1}{4}$
Now, $x_0=1,x_1=1+\frac{1}{4},x_2=1+2\times \frac{1}{4},x_3=1+3\times \frac{1}{4},x_4=1+4\times \frac{1}{4}$
i.e, $x_0=1,x_1=1.25,x_2=1.5,x_3=1.75,x_4=2$
$\Rightarrow y_0=1,y_1=0.8,y_2=0.667,y_3=0.571,y_4=0.5$
$\therefore$ Using Simpson's $\frac{1}{3}$rd rule
${\int }_1^2\frac{dx}{x}=\frac{1}{12}[(1+0.5)+4(0.8+0.571)+2\left(0.667\right)]$
$=\frac{1}{12}[1.5+5.484+1.334]$
$=\frac{1}{12}\left[8.318\right]=0.6932$