The problem requires us to approximate the integral \(\int_0^6 \frac{dx}{1+x^2}\) using the trapezoidal rule. Let's go through the steps involved in solving this using the method.
Trapezoidal Rule Formula:
The trapezoidal rule for approximate integration is given by:
\(T_n = \frac{b-a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]\)
Here, \(a\) and \(b\) are the limits of integration, \(n\) is the number of subintervals, and \(f(x_i)\) are the function values at each subinterval.
Step 1: Defining the function:
The function to integrate is \(f(x) = \frac{1}{1+x^2}\).
Step 2: Setting up parameters:
Given limits are \(a=0\), \(b=6\). Let's use 6 subintervals (n=6) for simplicity.
Step 3: Calculating the width of each interval:
Width, \(h\), is given by:
\(h = \frac{b-a}{n} = \frac{6-0}{6} = 1\)
Step 4: Compute the function values at each subinterval:
Step 5: Applying the trapezoidal rule:
\(T_6 = \frac{1}{2} \left[ 1 + 2 \times \left( \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} + \frac{1}{26} \right) + \frac{1}{37} \right]\)
\(= \frac{1}{2} \left[ 1 + 2 \times \left( 0.5 + 0.2 + 0.1 + 0.0588 + 0.0385 \right) + 0.027 \right]\)
\(= \frac{1}{2} \left[ 1 + 1.796 + 0.027 \right]\)
\(= \frac{1}{2} \times 2.823 = 1.4115\)
The computed approximate value is close to 1.4108, which is one of the given options.
Thus, the correct answer is 1.4108.