357. Calculate the integral $\int_{2}^{12} \frac{d x}{x}$ using the trapezoidal rule, dividing the interval of integration into 10 equal parts. Estimate the error.
Solution. According to the condition, $a=2, b=12, n=10, \Delta x=1$. We will calculate the values of the integrand function $y=1 / x$ at the corresponding division points: $x_{0}=2, \quad y_{0}=1 / 2=0.5 ; \quad x_{1}=3, y_{1}=1 / 3 \approx 0.3333 ; \quad x_{2}=4, y_{2}=$
$=1 / 4=0.25 ; x_{3}=5, y_{3}=1 / 5=0.2 ; x_{4}=6, y_{4}=1 / 6 \approx 0.1667 ; x_{5}=7, y_{5}=$ $=1 / 7 \approx 0.1429 ; \quad x_{6}=8, \quad y_{6}=0.125 ; \quad x_{7}=9, \quad y_{7} \approx 0.1111 ; \quad x_{8}=10, \quad y_{8}=0.1 ;$ $x_{9}=11, y_{9} \approx 0.0909 ; x_{10}=12, y_{10} \approx 0.0833$.

Substituting the found values into formula (12), we get $\int_{2}^{12} \frac{d x}{x} \approx 1\left(\frac{0.5+0.0833}{2}+0.3333+0.25+0.2+0.1667+0.1429+0.125+\right.$

$$
+0.1111+0.1+0.0909) \approx 1.812
$$

We will calculate this integral using the Newton-Leibniz formula:

$$
\int_{2}^{12} \frac{d x}{x}=\left.\ln x\right|_{2} ^{12}=\ln 12-\ln 2=\ln 6 \approx 1.792
$$

We will find the error in the approximate calculation:

$$
\delta=\frac{1.812-1.792}{1.792} \cdot 100 \% \approx 1.1 \%
$$