Trapezoidal, midpoint and Simpson's rules

Problem 4.381 · medium

1. \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]
   Δx = (b − a)/n.✓ Proved
2. The midpoint rule uses the points x = 1/8, 3/8, 5/8, 7/8 with weights 1, 1, 1, 1, all times 1/4.
3. \[ \frac{1 \cdot 256 \cdot \frac{1}{113} + 1 \cdot 256 \cdot \frac{1}{89} + 1 \cdot 256 \cdot \frac{1}{73} + 1 \cdot 256 \cdot \frac{1}{65}}{4} = \frac{150166784}{47720465} \]
   Weighted sum of the function values.✓ Proved
4. \[ \int\limits_{0}^{1} \frac{4}{x^{2} + 1}\, dx = \pi \]
   The exact value, ≈ 3.141593, for comparison: the error is about 5.21e-03.✓ Proved