Integrals of powers of sine and cosine
Problem 4.714 · easy
Evaluate \( \displaystyle \int_0^{\pi} \sin^{2}{\left(x \right)}\, dx \).
- Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, then integrate.
- \[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}\right) = \sin^{2}{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
- \[ \frac{\pi}{2} \]Evaluate from 0.✓ Proved
Answer \( \frac{\pi}{2} \)
Lines: 2 proved, 1 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 | ✓ 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 | numerical quadrature |
Reviewers
gpt-oss:20b: fail (misleading) — The first sentence incorrectly references cos²x and claims both powers are even, which is irrelevant to the problem. It also does not describe the actual integration steps, misleading a student about the method used.qwen3.6:27b-mlx: fail (misleading) — The solution claims 'Both powers are even', but the integrand is sin^2(x), which involves only one function. This phrasing is confusing and suggests a misunderstanding of the problem structure (e.g., implying a product like sin^2(x)cos^2(x)).
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-08 — The first sentence incorrectly references cos²x and claims both powers are even, which is irrelevant to the problem. It also does not describe the actual integration steps, misleading a student about the method used.qwen3.6:27b-mlx: fail (misleading) 2026-10-08 — The solution claims 'Both powers are even', but the integrand is sin^2(x), which involves only one function. This phrasing is confusing and suggests a misunderstanding of the problem structure (e.g., implying a product like sin^2(x)cos^2(x)).gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution text explicitly mentions using the identity for cos²x, which is irrelevant to the problem involving only sin²x. This indicates a copy-paste error or confusion in the setup description.
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_powers_integral, checked 2026-10-08 with SymPy 1.14.0.