Use the trapezoidal rule with $n=5$ to approximate
\[
\int_{1}^{6} \frac{\cos (x)}{x} d x
\]
Answer
Substitute these values into the trapezoidal rule formula: \[\int_{1}^{6} \frac{\cos (x)}{x} dx \approx \frac{1}{2} [\frac{\cos (1)}{1} + 2*\frac{\cos (2)}{2} + 2*\frac{\cos (3)}{3} + 2*\frac{\cos (4)}{4} + 2*\frac{\cos (5)}{5} + \frac{\cos (6)}{6}]\]
Step 1 ：Calculate the width of each subinterval: \(h = \frac{b - a}{n} = \frac{6 - 1}{5} = 1\)
Step 2 ：Calculate the \(x_i\) values: \(x_0 = 1 + 0*1 = 1\), \(x_1 = 1 + 1*1 = 2\), \(x_2 = 1 + 2*1 = 3\), \(x_3 = 1 + 3*1 = 4\), \(x_4 = 1 + 4*1 = 5\), \(x_5 = 1 + 5*1 = 6\)
Step 3 ：Substitute these \(x_i\) values into the function \(f(x) = \frac{\cos (x)}{x}\): \(f(x_0) = \frac{\cos (1)}{1}\), \(f(x_1) = \frac{\cos (2)}{2}\), \(f(x_2) = \frac{\cos (3)}{3}\), \(f(x_3) = \frac{\cos (4)}{4}\), \(f(x_4) = \frac{\cos (5)}{5}\), \(f(x_5) = \frac{\cos (6)}{6}\)
Step 4 ：Substitute these values into the trapezoidal rule formula: \[\int_{1}^{6} \frac{\cos (x)}{x} dx \approx \frac{1}{2} [\frac{\cos (1)}{1} + 2*\frac{\cos (2)}{2} + 2*\frac{\cos (3)}{3} + 2*\frac{\cos (4)}{4} + 2*\frac{\cos (5)}{5} + \frac{\cos (6)}{6}]\]
