Trigonometric substitution
Problem 4.573 · hard
Evaluate \( \displaystyle \int \frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}}\, dx \).
- 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.
- \[ \frac{d}{d x} \frac{x}{\sqrt{1 - x^{2}}} = \frac{1}{\left(1 - x^{2}\right)^{\frac{3}{2}}} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{x}{\sqrt{1 - x^{2}}} + C \)
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: passqwen3.6:27b-mlx: fail (error) — The substitution x = sin(theta) is missing the coefficient '1' in the text description (written as 'x = 1 sin theta' which is syntactically odd but mathematically okay, however, the standard form is x = sin(theta)). More critically, the solution skips the actual integration steps and the back-substitution process, jumping directly to a verification step. While the verification proves the answer is correct, the 'solution' does not actually solve the integral as requested; it merely verifies a given answer. A solution must demonstrate the derivation. Furthermore, the sentence 'Rewrite the integral in theta, integrate, then use a right triangle...' is a plan, not a step, and the subsequent line is a check, not the execution of that plan. This fails to provide the requested evaluation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The substitution x = sin(theta) is missing the coefficient '1' in the text description (written as 'x = 1 sin theta' which is syntactically odd but mathematically okay, however, the standard form is x = sin(theta)). More critically, the solution skips the actual integration steps and the back-substitution process, jumping directly to a verification step. While the verification proves the answer is correct, the 'solution' does not actually solve the integral as requested; it merely verifies a given answer. A solution must demonstrate the derivation. Furthermore, the sentence 'Rewrite the integral in theta, integrate, then use a right triangle...' is a plan, not a step, and the subsequent line is a check, not the execution of that plan. This fails to provide the requested evaluation.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The substitution in line 1 is syntactically invalid ('x = 1 sin theta' is not standard notation for x = sin(theta)). Furthermore, the solution skips the crucial step of computing the differential dx = cos(theta) d(theta), which is necessary to transform the integral correctly.
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 2026-10-07 with SymPy 1.14.0.