Integrals giving inverse trig functions
Problem 4.635 · medium
Evaluate \( \displaystyle \int \frac{2}{x^{2} + 2 x + 5}\, dx \).
- \[ x^{2} + 2 x + 5 = \left(x + 1\right)^{2} + 4 \]Complete the square.✓ Proved
- With u = x + 1 and a = 2, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)} = \frac{2}{x^{2} + 2 x + 5} \]An antiderivative is atan(x/2 + 1/2); differentiate to confirm.✓ Proved
Answer \( \operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)} + C \)
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: fail (error) — Sentence 2 incorrectly states the transformed integral. After completing the square, the integral becomes ∫2/(u²+4)du, not c·∫du/(u²+a²) with a=2. The missing factor 2 leads to an incorrect antiderivative unless the constant factor is explicitly handled.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Sentence 2 incorrectly states the transformed integral. After completing the square, the integral becomes ∫2/(u²+4)du, not c·∫du/(u²+a²) with a=2. The missing factor 2 leads to an incorrect antiderivative unless the constant factor is explicitly handled.qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution omits the constant factor 1/2 required by the standard integral formula ∫ du/(u² + a²) = (1/a) arctan(u/a). Consequently, the stated answer is off by a factor of 2.
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 2026-10-07 with SymPy 1.14.0.