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Trigonometric substitution

Problem 4.429 · hard

Evaluate \( \displaystyle \int \sqrt{1 - x^{2}}\, dx \).

The radical suggests x = 1 sin θ.
\[ 1 - \sin^{2}{\left(\theta \right)} = \cos^{2}{\left(\theta \right)} \]
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.
\[ \frac{d}{d x} \left(\frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2}\right) = \sqrt{1 - x^{2}} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2} + C \)

Lines: 2 proved, 2 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	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: pass
qwen3.6:27b-mlx: fail (error) — The substitution in line 1 is missing the coefficient '1' (written as 'x = 1 sin theta' instead of 'x = sin theta' or 'x = 1 \cdot \sin \theta'), which is syntactically awkward but mathematically acceptable. However, the primary issue is that line 3 claims to rewrite the integral in theta and integrate, but the solution skips the actual integration steps and the back-substitution process entirely, jumping straight to the verification in line 4. This fails to demonstrate the method suggested in line 1 and 3.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The substitution in line 1 is missing the coefficient '1' (written as 'x = 1 sin theta' instead of 'x = sin theta' or 'x = 1 \cdot \sin \theta'), which is syntactically awkward but mathematically acceptable. However, the primary issue is that line 3 claims to rewrite the integral in theta and integrate, but the solution skips the actual integration steps and the back-substitution process entirely, jumping straight to the verification in line 4. This fails to demonstrate the method suggested in line 1 and 3.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The substitution x = sin(theta) is missing the differential dx = cos(theta) dtheta, which is essential for the method described. The solution skips the actual integration steps and relies on a verification step that does not constitute a derivation.
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.