Integrals giving inverse trig functions
Problem 4.509 · medium
Evaluate \( \displaystyle \int_{-2}^{0} \frac{1}{x^{2} + 2 x + 2}\, dx \).
- \[ x^{2} + 2 x + 2 = \left(x + 1\right)^{2} + 1 \]Complete the square.✓ Proved
- With u = x + 1 and a = 1, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \operatorname{atan}{\left(x + 1 \right)} = \frac{1}{x^{2} + 2 x + 2} \]An antiderivative is atan(x + 1); differentiate to confirm.✓ Proved
- \[ \frac{\pi}{2} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{2} \)
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 change of limits when substituting u=x+1 and does not actually evaluate the definite integral; the final step merely states an identity rather than computing the antiderivative at the bounds.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and the final value. Although the intermediate sentence is informal and the limit evaluation step is opaque, the mathematical result is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The solution omits the change of limits when substituting u=x+1 and does not actually evaluate the definite integral; the final step merely states an identity rather than computing the antiderivative at the bounds.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the antiderivative and the final value. Although the intermediate sentence is informal and the limit evaluation step is opaque, the mathematical result is correct.gpt-oss:20b: fail (error) 2026-10-06 — The solution omits the change of limits after the substitution u=x+1 and incorrectly claims the definite integral equals π/2 without showing the evaluation. The final step is incomplete and does not justify the stated answer.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly completes the square, identifies the antiderivative, and evaluates the definite integral to the correct value.
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.