Integrals giving inverse trig functions
Problem 4.814 · medium
Evaluate \( \displaystyle \int_{3}^{3 + 4 \sqrt{3}} \frac{3}{x^{2} - 6 x + 25}\, dx \).
- \[ x^{2} - 6 x + 25 = \left(x - 3\right)^{2} + 16 \]Complete the square.✓ Proved
- With u = x − 3 and a = 4, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \frac{3 \operatorname{atan}{\left(\frac{x}{4} - \frac{3}{4} \right)}}{4} = \frac{3}{x^{2} - 6 x + 25} \]An antiderivative is 3*atan(x/4 - 3/4)/4; differentiate to confirm.✓ Proved
- \[ \frac{\pi}{4} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{4} \)
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’s sentences are incomplete and incorrect. Sentence 2 claims the integral becomes “c·∫ du/(u²+a²)” without specifying the constant c or showing how the limits transform. Sentence 4 merely repeats the answer without actually evaluating the antiderivative at the bounds. These omissions mean the reasoning is not sound and would mislead a student.qwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the antiderivative. The final evaluation is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-10 — The solution’s sentences are incomplete and incorrect. Sentence 2 claims the integral becomes “c·∫ du/(u²+a²)” without specifying the constant c or showing how the limits transform. Sentence 4 merely repeats the answer without actually evaluating the antiderivative at the bounds. These omissions mean the reasoning is not sound and would mislead a student.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the antiderivative. The final evaluation is correct.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly completes the square, identifies the standard arctangent integral form, and evaluates the definite integral to the correct result.
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.