Integrals giving inverse trig functions
Problem 4.959 · medium
Evaluate \( \displaystyle \int_{2}^{\sqrt{3} + 2} \frac{3}{x^{2} - 4 x + 5}\, dx \).
- \[ x^{2} - 4 x + 5 = \left(x - 2\right)^{2} + 1 \]Complete the square.✓ Proved
- With u = x − 2 and a = 1, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} 3 \operatorname{atan}{\left(x - 2 \right)} = \frac{3}{x^{2} - 4 x + 5} \]An antiderivative is 3*atan(x - 2); differentiate to confirm.✓ Proved
- \[ \pi \]Evaluate between the limits.✓ Proved
Answer \( \pi \)
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 (error) — The solution omits the factor 3 in the substitution step and provides no evaluation of the definite integral; the final statement "pi = pi" is meaningless. The setup is correct, but the reasoning is incomplete and incorrect.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and evaluates the definite integral to get the correct result. The steps are logically sound and algebraically verified.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution omits the factor 3 in the substitution step and provides no evaluation of the definite integral; the final statement "pi = pi" is meaningless. The setup is correct, but the reasoning is incomplete and incorrect.qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the antiderivative and evaluates the definite integral to get the correct result. The steps are logically sound and algebraically verified.gpt-oss:20b: fail (error) 2026-10-12 — The solution’s sentences do not correctly set up or evaluate the integral. Sentence 2 incorrectly omits the constant factor 3 and fails to transform the limits; sentence 4 provides no evaluation of the definite integral and merely states “pi = pi.”qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly completes the square, identifies the standard arctangent integral form, and evaluates the definite integral to obtain the correct result of pi.
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.