Integrals giving inverse trig functions
Problem 4.639 · medium
Evaluate \( \displaystyle \int_{-3}^{-3 + \frac{5 \sqrt{2}}{2}} \frac{3}{\sqrt{- x^{2} - 6 x + 16}}\, dx \).
- \[ - x^{2} - 6 x + 16 = 25 - \left(x + 3\right)^{2} \]Complete the square.✓ Proved
- With u = x + 3 and a = 5, this is c·∫ du/√(a² − u²)
- \[ \frac{d}{d x} 3 \operatorname{asin}{\left(\frac{x}{5} + \frac{3}{5} \right)} = \frac{3}{\sqrt{- x^{2} - 6 x + 16}} \]An antiderivative is 3*asin(x/5 + 3/5); differentiate to confirm.✓ Proved
- \[ \frac{3 \pi}{4} \]Evaluate between the limits.✓ Proved
Answer \( \frac{3 \pi}{4} \)
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 (misleading) — Sentence 2 incorrectly states the integral form: it omits the factor 3 and does not transform the limits. Sentence 4 asserts the final value without any calculation, so the conclusion is unsupported.qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative via completing the square and substitution. The final evaluation is correct, as the limits map to arcsin(0) and arcsin(1/√2), yielding 3π/4.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-07 — Sentence 2 incorrectly states the integral form: it omits the factor 3 and does not transform the limits. Sentence 4 asserts the final value without any calculation, so the conclusion is unsupported.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the antiderivative via completing the square and substitution. The final evaluation is correct, as the limits map to arcsin(0) and arcsin(1/√2), yielding 3π/4.gpt-oss:20b: fail (error) 2026-10-07 — Sentence 2 incorrectly introduces an unspecified constant ‘c’ and does not transform the limits of integration. The setup omits the necessary change of variables step and the resulting definite integral, which would mislead a student about how to apply the substitution and evaluate the bounds.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the antiderivative via completing the square and substitution, and the evaluation of the definite integral 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 2026-10-07 with SymPy 1.14.0.