Trigonometric substitution
Problem 4.426 · hard
Evaluate \( \displaystyle \int \frac{x^{2}}{\sqrt{1 - x^{2}}}\, dx \).
- 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.
- \[ \frac{d}{d x} \left(- \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2}\right) = \frac{x^{2}}{\sqrt{1 - x^{2}}} \]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: passqwen3.6:27b-mlx: fail (error) — The substitution in step 1 is syntactically incorrect ('x = 1 sin theta' is not standard notation and likely implies a missing operator or typo). Furthermore, the solution skips the actual integration steps, jumping from setup to a verification of the final answer, which fails to demonstrate the method requested by the problem statement.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The substitution in step 1 is syntactically incorrect ('x = 1 sin theta' is not standard notation and likely implies a missing operator or typo). Furthermore, the solution skips the actual integration steps, jumping from setup to a verification of the final answer, which fails to demonstrate the method requested by the problem statement.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — Line 1 contains a syntax error ('x = 1 sin θ' is not valid mathematical notation) and fails to specify the necessary bounds for θ (e.g., -π/2 ≤ θ ≤ π/2) to ensure the square root simplifies correctly to cos θ rather than |cos θ|. Line 4 verifies the final answer but does not constitute a valid derivation of the 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.