Integrals giving inverse trig functions
Problem 4.503 · medium
Evaluate \( \displaystyle \int \frac{3}{x^{2} - 2 x + 26}\, dx \).
- \[ x^{2} - 2 x + 26 = \left(x - 1\right)^{2} + 25 \]Complete the square.✓ Proved
- With u = x − 1 and a = 5, this is c·∫ du/(u² + a²)Reviewed
- \[ \frac{d}{d x} \frac{3 \operatorname{atan}{\left(\frac{x}{5} - \frac{1}{5} \right)}}{5} = \frac{3}{x^{2} - 2 x + 26} \]An antiderivative is 3*atan(x/5 - 1/5)/5; differentiate to confirm.✓ Proved
Answer \( \frac{3 \operatorname{atan}{\left(\frac{x}{5} - \frac{1}{5} \right)}}{5} + C \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result via differentiation. The logic is sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result via differentiation. The logic is sound.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly completes the square and applies the standard arctangent integration formula. The final verification confirms the 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 2026-10-06 with SymPy 1.14.0.