Integrals giving inverse trig functions
Problem 4.960 · medium
Evaluate \( \displaystyle \int_{-1}^{0} \frac{1}{x^{2} + 2 x + 2}\, dx \).
- \[ x^{2} + 2 x + 2 = \left(x + 1\right)^{2} + 1 \]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
- \[ \frac{\pi}{4} \]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 skips the change of limits when substituting u=x+1 and does not actually evaluate the antiderivative at the bounds; it merely states the result without justification.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and evaluates the definite integral. The intermediate sentence is informal but mathematically sound in its implication.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution skips the change of limits when substituting u=x+1 and does not actually evaluate the antiderivative at the bounds; it merely states the result without justification.qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the antiderivative and evaluates the definite integral. The intermediate sentence is informal but mathematically sound in its implication.gpt-oss:20b: fail (error) 2026-10-12 — The solution omits the change of limits when substituting u=x+1 and does not actually evaluate the definite integral; the final statement "pi/4 = pi/4" is not a justification of the result.qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly completes the square, identifies the antiderivative as arctan(x+1), and evaluates the definite integral to pi/4. The steps are logically sound and algebraically correct.
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.