Integrals giving inverse trig functions
Problem 4.961 · medium
Evaluate \( \displaystyle \int_{1}^{1 + 3 \sqrt{3}} \frac{3}{x^{2} - 2 x + 10}\, dx \).
- \[ x^{2} - 2 x + 10 = \left(x - 1\right)^{2} + 9 \]Complete the square.✓ Proved
- With u = x − 1 and a = 3, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \operatorname{atan}{\left(\frac{x}{3} - \frac{1}{3} \right)} = \frac{3}{x^{2} - 2 x + 10} \]An antiderivative is atan(x/3 - 1/3); differentiate to confirm.✓ Proved
- \[ \frac{\pi}{3} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{3} \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The antiderivative in step 3 is missing a factor of 1/3. The derivative of atan((x-1)/3) is 1/(3((x-1)^2/9 + 1)) = 3/((x-1)^2 + 9), which matches th
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The antiderivative in step 3 is missing a factor of 1/3. The derivative of atan((x-1)/3) is 1/(3((x-1)^2/9 + 1)) = 3/((x-1)^2 + 9), which matches thgpt-oss:20b: inconclusive 2026-10-12 — reviewer response could not be parsed: {"verdict":"pass","severity":"","notes":"The solution’s sentences are correct, though the wording in sentence 2 could be clearer by explicitly stating the constant factor \(c=3\) and mentioning the trqwen3.6:27b-mlx: fail (error) 2026-10-12 — The antiderivative in line 3 is missing the necessary factor of 1/3. The integral of 3/(u^2 + 9) is atan(u/3), not 3*atan(u/3) or just atan(u/3) without adjusting the coefficient. Specifically, d/dx(atan((x-1)/3)) = 1/3 * 1/(((x-1)/3)^2 + 1) = 3/((x-1)^2 + 9). So the antiderivative is indeed atan((x-1)/3). However, line 2 suggests the form c * integral, implying a constant multiplier, but line 3 presents the antiderivative as just atan(x/3 - 1/3). Let's check the evaluation. F(x) = atan((x-1)/3). F(1+3sqrt(3)) = atan(3sqrt(3)/3) = atan(sqrt(3)) = pi/3. F(1) = atan(0) = 0. Result pi/3. The math is actually correct. Wait, let me re-read line 3 carefully. 'An antiderivative is atan(x/3 - 1/3)'. Derivative of atan((x-1)/3) is 1/(1 + ((x-1)/3)^2) * (1/3) = 1/3 * 1/(1 + (x-1)^2/9) = 1/3 * 9/(9 + (x-1)^2) = 3/((x-1)^2 + 9). This matches the integrand. So the antiderivative is correct. The evaluation is correct. Why did I think it was wrong? Ah, standard integral is 1/a atan(u/a). Here a=3. So 1/3 atan(u/3). But we have a factor of 3 in the numerator. 3 * (1/3 atan(u/3)) = atan(u/3). So the antiderivative is correct. The solution is sound.
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 2026-10-12 with SymPy 1.14.0.