Integrals giving inverse trig functions
Problem 4.955 · medium
Evaluate \( \displaystyle \int_{- \frac{5}{2}}^{- \frac{3}{2}} \frac{4}{\sqrt{- x^{2} - 4 x - 3}}\, dx \).
- \[ - x^{2} - 4 x - 3 = 1 - \left(x + 2\right)^{2} \]Complete the square.✓ Proved
- With u = x + 2 and a = 1, this is c·∫ du/√(a² − u²)Reviewed
- \[ \frac{d}{d x} 4 \operatorname{asin}{\left(x + 2 \right)} = \frac{4}{\sqrt{- x^{2} - 4 x - 3}} \]An antiderivative is 4*asin(x + 2); differentiate to confirm.✓ Proved
- \[ \frac{4 \pi}{3} \]Evaluate between the limits.✓ Proved
Answer \( \frac{4 \pi}{3} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative and evaluates it at the bounds. Although the intermediate sentence is informal, the final result is correct and the steps are logically sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly identifies the antiderivative and evaluates it at the bounds. Although the intermediate sentence is informal, the final result is correct and the steps are logically sound.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly completes the square, identifies the standard arcsine integral form, and evaluates the definite integral with the correct limits and result.
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.