Integrals giving inverse trig functions
Problem 4.502 · medium
Evaluate \( \displaystyle \int_{7}^{2 + 5 \sqrt{3}} \frac{1}{x^{2} - 4 x + 29}\, dx \).
- \[ x^{2} - 4 x + 29 = \left(x - 2\right)^{2} + 25 \]Complete the square.✓ Proved
- With u = x − 2 and a = 5, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \frac{\operatorname{atan}{\left(\frac{x}{5} - \frac{2}{5} \right)}}{5} = \frac{1}{x^{2} - 4 x + 29} \]An antiderivative is atan(x/5 - 2/5)/5; differentiate to confirm.✓ Proved
- \[ \frac{\pi}{60} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{60} \)
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 | numerical quadrature |
Reviewers
gpt-oss:20b: fail (style) — [domain objection, downgraded to style] The solution omits the evaluation of the definite integral: it never substitutes the limits 7 and 2+5√3 into the antiderivative, so the claimed result π/60 is unsupported. Additionally, the sentence in step 2 introduces an undefined constant c, which is unnecessary and could confuse a student.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative via completing the square and confirms it by differentiation. The final evaluation is correct.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-06 — [domain objection, downgraded to style] The solution omits the evaluation of the definite integral: it never substitutes the limits 7 and 2+5√3 into the antiderivative, so the claimed result π/60 is unsupported. Additionally, the sentence in step 2 introduces an undefined constant c, which is unnecessary and could confuse a student.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the antiderivative via completing the square and confirms it by differentiation. The final evaluation is correct.gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claiming the integral becomes a constant times ∫du/(u²+a²) is unclear and omits the fact that the integrand is exactly 1/(u²+25); the subsequent evaluation step is also missing, so a student would not learn how to compute the definite integral.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the antiderivative. The final evaluation is correct.
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/inverse_trig_integral, checked 2026-10-06 with SymPy 1.14.0.