# How do you use the trapezoidal rule to find the integral from 1 to 4 for 6sqrt(lnx) with n=6?

Dec 15, 2016

${\int}_{1}^{4} 6 \sqrt{\ln x} \mathrm{dx} \approx 15.54800 \text{ } \left(5 \mathrm{dp}\right)$

#### Explanation:

The values of $y = 6 \sqrt{\ln x}$ are tabulated as follows (using Excel) working to 5dp

Using the trapezoidal rule:

${\int}_{a}^{b} y \mathrm{dx} \approx \frac{h}{2} \left\{\left({y}_{0} + {y}_{n}\right) + 2 \left({y}_{1} + {y}_{2} + \ldots + {y}_{n - 1}\right)\right\}$

We have:

 int_(1)^(4) 6sqrt(lnx)dx ~~ 0.5/2 { 0 + 7.06446 + 2(3.82057 + 4.99533 + 5.74338 + 6.28888 + 6.71561)}
$\text{ } = 0.25 \left\{7.06446 + 2 \left(27.56378\right)\right\}$
$\text{ } = 0.25 \left\{7.06446 + 55.12755\right\}$
$\text{ } = 0.25 \left\{62.19201\right\}$
$\text{ } = 15.54800$