Integrals giving inverse trig functions
Problem 4.808 · medium
Evaluate \( \displaystyle \int_{-2}^{-2 + \frac{5 \sqrt{2}}{2}} \frac{2}{\sqrt{- x^{2} - 4 x + 21}}\, dx \).
- \[ - x^{2} - 4 x + 21 = 25 - \left(x + 2\right)^{2} \]Complete the square.✓ Proved
- With u = x + 2 and a = 5, this is c·∫ du/√(a² − u²)
- \[ \frac{d}{d x} 2 \operatorname{asin}{\left(\frac{x}{5} + \frac{2}{5} \right)} = \frac{2}{\sqrt{- x^{2} - 4 x + 21}} \]An antiderivative is 2*asin(x/5 + 2/5); 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 (misleading) — The solution’s sentences omit key details: the constant factor in the transformed integral is not stated, and the evaluation of the definite integral is glossed over with a trivial equality. A student could be misled into thinking the steps are fully justified when they are not.qwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard arcsine integral form, and verifies the antiderivative. The final evaluation is correct.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-10 — The solution’s sentences omit key details: the constant factor in the transformed integral is not stated, and the evaluation of the definite integral is glossed over with a trivial equality. A student could be misled into thinking the steps are fully justified when they are not.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly completes the square, identifies the standard arcsine integral form, and verifies the antiderivative. The final evaluation is correct.gpt-oss:20b: fail (error) 2026-10-10 — The solution’s second sentence incorrectly states the transformed integral as a generic c·∫du/√(a²−u²) without specifying the correct integrand 2/√(a²−u²) or showing the change of limits. Additionally, the final evaluation step merely repeats π/2 without demonstrating how the definite integral yields that value. These omissions mean the setup and conclusion are incomplete and potentially misleading.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly completes the square, identifies the standard arcsine integral form, and verifies the antiderivative. The final evaluation is 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 2026-10-10 with SymPy 1.14.0.