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Integrals of powers of sine and cosine

Problem 4.410 · medium

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

The power of sine is odd: keep one sin x for du, write the rest as (1 − cos²x)^0, and let u = cos x (du = −sin x dx).
\[ \frac{d}{d u} \left(- \frac{u^{5}}{5}\right) = - u^{4} \]
∫ -u**4 du = -u**5/5.✓ Proved
\[ \frac{d}{d x} \left(- \frac{\cos^{5}{\left(x \right)}}{5}\right) = \sin{\left(x \right)} \cos^{4}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer $ - \frac{\cos^{5}{\left(x \right)}}{5} + C $

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	difference quotients of the answer match the integrand

Reviewers

gpt-oss:20b: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The sentence claims to rewrite the remaining factor as $(1-\cos^2x)^0$, which is incorrect because the remaining factor is $\cos^4x$, not $1$. This m
qwen3.6:27b-mlx: pass — The solution correctly identifies the strategy for integrating odd powers of sine and verifies the result via differentiation. The algebraic steps are sound.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the strategy for integrating odd powers of sine and verifies the result via differentiation. The algebraic steps are sound.
gpt-oss:20b: inconclusive 2026-10-05 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The sentence claims to rewrite the remaining factor as $(1-\cos^2x)^0$, which is incorrect because the remaining factor is $\cos^4x$, not $1$. This m
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the strategy for integrating powers of sine and cosine where the power of sine is odd. The substitution and final result are correct.
gpt-oss:20b: fail (misleading) 2026-10-05 — The first sentence incorrectly states that the remaining factor is (1−cos²x)^0, which is not derived from the integrand. While the substitution u=cos x is correct, the description of the power‑reduction step is misleading and could confuse a student.

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.