Use the trapezium rule with four sub-intervals to estimate: \[ \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\sin x} d x \text {; } \] Give your answer correct th three decimal places:

To estimate the integral \[ \int_{0}^{\frac{\pi}{2}} \frac{1}{1+\sin x} \, dx \] using the trapezium rule with four sub-intervals, we first need to determine the width of each sub-interval. 1. **Determine the interval and width**: The interval is from \(0\) to \(\frac{\pi}{2}\), which has a length of \[ b - a = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \] With \(n = 4\) sub-intervals, the width \(h\) of each sub-interval is \[ h = \frac{b - a}{n} = \frac{\frac{\pi}{2}}{4} = \frac{\pi}{8}. \] 2. **Determine the x-values**: The x-values at the endpoints of the sub-intervals are: \[ x_0 = 0, \] \[ x_1 = 0 + h = \frac{\pi}{8}, \] \[ x_2 = \frac{\pi}{8} + h = \frac{\pi}{4}, \] \[ x_3 = \frac{\pi}{4} + h = \frac{3\pi}{8}, \] \[ x_4 = \frac{3\pi}{8} + h = \frac{\pi}{2}. \] 3. **Evaluate the function at these points**: We need to evaluate the function \[ f(x) = \frac{1}{1 + \sin x} \] at each of these points: - \(f(x_0) = f(0) = \frac{1}{1 + \sin(0)} = \frac{1}{1 + 0} = 1\) - \(f(x_1) = f\left(\frac{\pi}{8}\right) = \frac{1}{1 + \sin\left(\frac{\pi}{8}\right)}\) - \(f(x_2) = f\left(\frac{\pi}{4}\right) = \frac{1}{1 + \sin\left(\frac{\pi}{4}\right)} = \frac{1}{1 + \frac{\sqrt{2}}{2}} = \frac{2}{2 + \sqrt{2}}\) - \(f(x_3) = f\left(\frac{3\pi}{8}\right) = \frac{1}{1 + \sin\left(\frac{3\pi}{8}\right)}\) - \(f(x_4) = f\left(\frac{\pi}{2}\right) = \frac{1}{1 + \sin\left(\frac{\pi}{2}\right)} = \frac{1}{1 + 1} = \frac{1}{2}\) We can calculate \(f\left(\frac{\pi}{8}\right)\) and \(f\left(\frac{3\pi}{8}\right)\) using the sine values: - \(\sin\left(\frac{\pi}{8}\right) = \frac{\sqrt{2 - \sqrt{2}}}{2}\) - \(\sin\left(\frac{3\pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2}\) Thus, \[ f\left(\frac{\pi}{8}\right) = \frac{1}{1 + \frac{\sqrt{2 - \sqrt{2}}}{2}} = \frac{2}{2 + \sqrt{2 - \sqrt{2}}} \] and \[ f\left(\frac{3\pi}{8}\right) = \frac{1}{1 + \frac{\sqrt{2 + \sqrt{2}}}{2}} = \frac{2}{2 + \sqrt{2 + \sqrt{2}}}. \] 4. **Calculate the trapezium rule**: The trapezium rule formula is given by: \[ \int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \right). \] Plugging in the values: \[ \int_