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Integrals of powers of sine and cosine

Problem 4.857 · medium

Evaluate \( \displaystyle \int_0^{\frac{\pi}{4}} \sin^{2}{\left(x \right)}\, dx \).

Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, then integrate.
Reviewed
\[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}\right) = \sin^{2}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved
\[ - \frac{1}{4} + \frac{\pi}{8} \]
Evaluate from 0.✓ Proved

Answer \( - \frac{1}{4} + \frac{\pi}{8} \)

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the power-reduction formula for sin²(x) and verifies the result via differentiation and evaluation. The logic is sound.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the power-reduction formula for sin²(x) and verifies the result via differentiation and evaluation. The logic is sound.
gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (misleading) 2026-10-10 — 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)).

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 2026-10-10 with SymPy 1.14.0.