Integrals of powers of sine and cosine
Problem 4.860 · medium
Evaluate \( \displaystyle \int_0^{\frac{\pi}{2}} \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx \).
- The power of cosine is odd: keep one cos x for du, write the rest as (1 − sin²x)^0, and let u = sin x.
- \[ \frac{d}{d u} \frac{u^{5}}{5} = u^{4} \]∫ u**4 du = u**5/5.✓ Proved
- \[ \frac{d}{d x} \frac{\sin^{5}{\left(x \right)}}{5} = \sin^{4}{\left(x \right)} \cos{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
- \[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]Evaluate from 0.✓ Proved
Answer \( \frac{1}{5} \)
Lines: 3 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 |
| 4 | ✓ 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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The first sentence incorrectly describes the substitution: it claims to write the remaining factor as $(1-\sin^2x)^0$ and misstates the role of the odd coqwen3.6:27b-mlx: pass — The solution correctly identifies the substitution method for an odd power of cosine, performs the integration and differentiation checks correctly, and evaluates the definite integral accurately.
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-10 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The first sentence incorrectly describes the substitution: it claims to write the remaining factor as $(1-\sin^2x)^0$ and misstates the role of the odd coqwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the substitution method for an odd power of cosine, performs the integration and differentiation checks correctly, and evaluates the definite integral accurately.gpt-oss:20b: fail (error) 2026-10-10 — Sentence 1 incorrectly rewrites sin⁴x as (1−sin²x)⁰ and claims to keep a cos x for du. The correct substitution is u=sin x, du=cos x dx, yielding ∫u⁴du without any extra factor. The statement misrepresents the setup and would mislead a student.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the substitution method for an odd power of cosine, executes the integration and boundary evaluation accurately, and verifies the result via 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 2026-10-10 with SymPy 1.14.0.