Trigonometric substitution
Problem 4.422 · hard
Evaluate \( \displaystyle \int \sqrt{16 - x^{2}}\, dx \).
- The radical suggests x = 4 sin θ.Reviewed
- \[ 16 - 16 \sin^{2}{\left(\theta \right)} = 16 \cos^{2}{\left(\theta \right)} \]The identity that clears the square root.✓ Proved
- Rewrite the integral in θ, integrate, then use a right triangle to express the trig functions of θ in terms of x.Reviewed
- \[ \frac{d}{d x} \left(\frac{x \sqrt{16 - x^{2}}}{2} + 8 \operatorname{asin}{\left(\frac{x}{4} \right)}\right) = \sqrt{16 - x^{2}} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{x \sqrt{16 - x^{2}}}{2} + 8 \operatorname{asin}{\left(\frac{x}{4} \right)} + C \)
✓ Nihil obstat Lines: 2 proved, 2 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 | F(b) − F(a) matches numerical quadrature of the integrand on a sub-interval of the domain |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the trigonometric substitution and verifies the result via differentiation. The steps are logically sound and mathematically accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the trigonometric substitution and verifies the result via differentiation. The steps are logically sound and mathematically accurate.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the trigonometric substitution and verifies the result via differentiation. The steps are logically sound and standard for this type of integral.gpt-oss:20b: pass 2026-10-05
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/trig_substitution, checked 2026-10-05 with SymPy 1.14.0.