Trapezoidal, midpoint and Simpson's rules
Problem 4.389 · medium
Approximate \( \displaystyle \int_{0}^{2} \frac{x}{x + 1}\, dx \) using the midpoint rule with \( \displaystyle n = 5 \).
- \[ \frac{2}{5} \]Δx = (b − a)/n.✓ Proved
- The midpoint rule uses the points x = 1/5, 3/5, 1, 7/5, 9/5 with weights 1, 1, 1, 1, 1, all times 2/5.
- \[ \frac{2 \left(1 \cdot 1 \cdot \frac{1}{6} + 1 \cdot 3 \cdot \frac{1}{8} + 1 \cdot 1 \cdot \frac{1}{2} + 1 \cdot 7 \cdot \frac{1}{12} + 1 \cdot 9 \cdot \frac{1}{14}\right)}{5} = \frac{127}{140} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{0}^{2} \frac{x}{x + 1}\, dx = 2 - \ln{\left(3 \right)} \]The exact value, ≈ 0.901388, for comparison: the error is about 5.76e-03.✓ Proved
Answer \( M_{5} = \frac{127}{140} \approx 0.907143 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the rule implemented separately in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/numerical_integration, checked 2026-10-05 with SymPy 1.14.0.