Use Simpson's rule with \( n=6 \) to approximate \( \int_{1}^{4} \frac{\cos (x)}{x} d x \) \( \square \)

To approximate the integral \( \int_{1}^{4} \frac{\cos(x)}{x} \, dx \) using Simpson's rule with \( n = 6 \), we first need to determine the width of each subinterval and the corresponding \( x \) values. ### Step 1: Determine the width of each subinterval The interval is from \( a = 1 \) to \( b = 4 \). The number of subintervals is \( n = 6 \). The width \( h \) of each subinterval is given by: \[ h = \frac{b - a}{n} = \frac{4 - 1}{6} = \frac{3}{6} = 0.5 \] ### Step 2: Determine the \( x \) values The \( x \) values at which we will evaluate the function are: \[ x_0 = 1, \quad x_1 = 1.5, \quad x_2 = 2, \quad x_3 = 2.5, \quad x_4 = 3, \quad x_5 = 3.5, \quad x_6 = 4 \] ### Step 3: Evaluate the function at these points We need to evaluate \( f(x) = \frac{\cos(x)}{x} \) at each of these points: \[ f(x_0) = f(1) = \frac{\cos(1)}{1} \approx 0.5403 \] \[ f(x_1) = f(1.5) = \frac{\cos(1.5)}{1.5} \approx 0.2837 \] \[ f(x_2) = f(2) = \frac{\cos(2)}{2} \approx -0.4161 \] \[ f(x_3) = f(2.5) = \frac{\cos(2.5)}{2.5} \approx -0.2473 \] \[ f(x_4) = f(3) = \frac{\cos(3)}{3} \approx -0.1543 \] \[ f(x_5) = f(3.5) = \frac{\cos(3.5)}{3.5} \approx -0.0869 \] \[ f(x_6) = f(4) = \frac{\cos(4)}{4} \approx -0.0579 \] ### Step 4: Apply Simpson's Rule Simpson's rule states: \[ \int_a^b f(x) \, dx \approx \frac{h}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6) \right) \] Substituting the values we calculated: \[ \int_1^4 f(x) \, dx \approx \frac{0.5}{3} \left( 0.5403 + 4(0.2837) + 2(-0.4161) + 4(-0.2473) + 2(-0.1543) + 4(-0.0869) + (-0.0579) \right) \] Calculating the weighted sum: \[ = \frac{0.5}{3} \left( 0.5403 + 1.1348 - 0.8322 - 0.9892 - 0.3086 - 0.3476 - 0.0579 \right) \] \[ = \frac{0.5}{3} \left( 0.5403 + 1.1348 - 0.8322 - 0.9892 - 0.3086 - 0.3476 - 0.0579 \right) \] \[ = \frac{0.5}{3} \left( -0.8604 \right) \] \[ = \frac{-0.4302}{3} \approx -0.1434 \] ### Final Result Thus, the approximate value of the integral \( \int_{1}^{4} \frac{\cos(x)}{x} \, dx \) using Simpson's rule with \( n = 6 \) is: \[ \boxed{-0.1434} \]