Value of ${\int }_4^{5.2}$ ln x dx using Simpson's one-third rule with interval size 0.3 is

The correct option is 1.83.

Step-by-Step Explanation:

We want to evaluate the definite integral:
$$I={\int }_4^{5.2}\ln xdx$$
using Simpson's one-third rule with an interval size of $h=0.3$.

1. Determine the number of intervals (n):
The lower limit is $a=4$ and the upper limit is $b=5.2$.
The number of intervals n is calculated as:
$$n=\frac{b-a}{h}=\frac{5.2-4}{0.3}=\frac{1.2}{0.3}=4$$
Since n = 4 is an even number, we can apply Simpson's one-third rule.

2. Calculate the grid points and the function values:
Let $y=f(x)=\ln x$. We find the value of y at each step $x_i=a+ih$ for i from 0 to 4:
• For i = 0: $x_0=4.0\to y_0=\ln (4.0)\approx 1.3863$
• For i = 1: