Home›Calculus 2›Integrals of powers of sine and cosine›Problem 4.418

Integrals of powers of sine and cosine

Problem 4.418 · medium

Evaluate \( \displaystyle \int \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx \).

Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, and sin x cos x = ½ sin 2x, then integrate.
Reviewed
\[ \frac{d}{d x} \left(\frac{x}{8} - \frac{\sin{\left(4 x \right)}}{32}\right) = \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved

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

Lines: 1 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
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: pass — The solution correctly identifies the strategy for handling even powers of sine and cosine using double-angle identities. The final answer is verified by differentiation.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the strategy for handling even powers of sine and cosine using double-angle identities. The final answer is verified by differentiation.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (misleading) 2026-10-05 — The solution lists three identities but only uses the first two; the third (sin x cos x = 1/2 sin 2x) is irrelevant to the method described and confusingly included. Additionally, the solution skips the actual integration steps, merely stating the identities and then verifying the final answer, which fails to demonstrate the derivation.
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_powers_integral, checked 2026-10-05 with SymPy 1.14.0.