Question

The value of the integral $\underset{0}{\overset{4}{\int }}\frac{dx}{1+x^2}$ obtained by using Trapezoidal rule with $h=1$ is

• A. $\frac{63}{85}$
• B. ${\tan }^{-1}$ (4)
• C. $\frac{108}{85}$
• D. $\frac{113}{85}$

Answer 1

Answer: (D)

Given integration is $\underset{0}{\overset{4}{\int }}\frac{dx}{1+x^2}$ and $h=1$.

$X$	0	1	2	3	4
$f(x)$	1	$\frac{1}{2}$	$\frac{1}{5}$	$\frac{1}{10}$	$\frac{1}{17}$

By using Trapezoidal rule,
$\underset{0}{\overset{4}{\int }}f(x)dx=\frac{h}{2}\left[\left(y_0+y_4\right)+2\left(y_1+y_2+y_3\right)\right]$
$=\frac{1}{2}\left[\left(1+\frac{1}{17}\right)+2\left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)\right]$
$=\frac{1}{2}\left[\frac{18}{17}+2\left(\frac{5+2+1}{10}\right)\right]$
$=\frac{1}{2}\left(\frac{18}{17}+\frac{8}{5}\right)=\frac{1}{2}\left(\frac{90+136}{85}\right)$
$=\frac{1}{2}\left(\frac{226}{85}\right)=\frac{113}{85}$