Integrals of powers of sine and cosine
Problem 4.1018 · medium
Evaluate \( \displaystyle \int_0^{\frac{\pi}{4}} \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx \).
- 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
- \[ \frac{\sqrt{2}}{40} \]Evaluate from 0.✓ Proved
Answer \( \frac{\sqrt{2}}{40} \)
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: fail (misleading) — The first sentence incorrectly describes the substitution: it claims to write the remaining factor as (1−sin²x)^0, which is 1, and does not correctly account for the sin⁴x term. This misleads a student about how to perform the u‑substitution and omits the change of limits.qwen3.6:27b-mlx: pass — The solution correctly identifies the substitution method for an odd power of cosine and verifies the result via differentiation and evaluation. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-13 — The first sentence incorrectly describes the substitution: it claims to write the remaining factor as (1−sin²x)^0, which is 1, and does not correctly account for the sin⁴x term. This misleads a student about how to perform the u‑substitution and omits the change of limits.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the substitution method for an odd power of cosine and verifies the result via differentiation and evaluation. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the substitution method for an odd power of cosine, performs the integration and evaluation accurately, and arrives at the correct final answer.
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 2026-10-13 with SymPy 1.14.0.