Q. The value of the integral $\int\limits_{0}^{4} \frac{d x}{1+x^{2}}$ obtained by using Trapezoidal rule with $h=1$ is
EAMCETEAMCET 2012
Solution:
Given integration is $\int\limits_{0}^{4} \frac{d x}{1+x^{2}}$ 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,
$ \int\limits_{0}^{4} f(x) d x =\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] $
$=\frac{1}{2}\left(\frac{18}{17}+\frac{8}{5}\right)=\frac{1}{2}\left(\frac{90+136}{85}\right)$
$=\frac{1}{2}\left(\frac{226}{85}\right)=\frac{113}{85}$
| $X$ | 0 | 1 | 2 | 3 | 4 |
| $f(x)$ | 1 | $\frac{1}{2}$ | $\frac{1}{5}$ | $\frac{1}{10}$ | $\frac{1}{17}$ |