Integrals giving inverse trig functions
Problem 4.815 · medium
Evaluate \( \displaystyle \int \frac{3}{x^{2} - 4 x + 20}\, dx \).
- \[ x^{2} - 4 x + 20 = \left(x - 2\right)^{2} + 16 \]Complete the square.✓ Proved
- With u = x − 2 and a = 4, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \frac{3 \operatorname{atan}{\left(\frac{x}{4} - \frac{1}{2} \right)}}{4} = \frac{3}{x^{2} - 4 x + 20} \]An antiderivative is 3*atan(x/4 - 1/2)/4; differentiate to confirm.✓ Proved
Answer \( \frac{3 \operatorname{atan}{\left(\frac{x}{4} - \frac{1}{2} \right)}}{4} + C \)
Lines: 2 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | difference quotient of the answer at x = -4.0 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution step is incorrect. With u = x - 2 and a = 4, the integral becomes 3 * ∫ du / (u^2 + 16). The standard integral ∫ du / (u^2 + a^2) i
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution step is incorrect. With u = x - 2 and a = 4, the integral becomes 3 * ∫ du / (u^2 + 16). The standard integral ∫ du / (u^2 + a^2) igpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims the integral is c * ∫ du/(u² + a²) but fails to specify that c = 3. Consequently, the transition from the integral form to the final answer lacks the necessary constant factor, making the derivation incomplete and logically disconnected.
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 2026-10-10 with SymPy 1.14.0.