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Trigonometric substitution

Problem 4.885 · hard

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

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.
\[ \frac{d}{d x} \left(- \frac{\sqrt{1 - x^{2}}}{x}\right) = \frac{1}{x^{2} \sqrt{1 - x^{2}}} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( - \frac{\sqrt{1 - x^{2}}}{x} + C \)

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 step 1 is written as 'x = 1 sin theta', which is syntactically awkward and potentially confusing, but the primary error is that the solution skips the actual integration steps. It claims to rewrite and integrate in step 3 but provides no work, jumping straight to a verification in step 4. While the final answer is correct, the solution fails to demonstrate the method requested by the setup, making it an incomplete and misleading guide for a student trying to learn the technique.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The substitution in step 1 is written as 'x = 1 sin theta', which is syntactically awkward and potentially confusing, but the primary error is that the solution skips the actual integration steps. It claims to rewrite and integrate in step 3 but provides no work, jumping straight to a verification in step 4. While the final answer is correct, the solution fails to demonstrate the method requested by the setup, making it an incomplete and misleading guide for a student trying to learn the technique.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The substitution is stated as 'x = 1 sin θ', which is syntactically incorrect and ambiguous (missing multiplication sign or clear function notation). Furthermore, the solution skips the actual integration steps and the back-substitution process, jumping directly to a verification step, which fails to demonstrate the method requested by the problem statement.
gpt-oss:20b: pass 2026-10-11

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 2026-10-11 with SymPy 1.14.0.