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Integrals of powers of sine and cosine

Problem 4.565 · medium

Evaluate \( \displaystyle \int \sin^{4}{\left(x \right)}\, dx \).

Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, then integrate.
\[ \frac{d}{d x} \left(\frac{3 x}{8} - \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32}\right) = \sin^{4}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \frac{3 x}{8} - \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32} + C \)

Lines: 1 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	difference quotients of the answer match the integrand

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The sentence claims 'Both powers are even', but the integrand sin^4(x) involves only one trigonometric function, not a product of sine and cosine. While the method described is valid for reducing the power, the premise is factually incorrect regarding the structure of the integrand.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The sentence claims 'Both powers are even', but the integrand sin^4(x) involves only one trigonometric function, not a product of sine and cosine. While the method described is valid for reducing the power, the premise is factually incorrect regarding the structure of the integrand.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the standard method for integrating even powers of sine using power-reduction formulas. The final answer is verified by differentiation.

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 2026-10-07 with SymPy 1.14.0.