Integrals of powers of sine and cosine
Problem 4.861 · easy
Evaluate \( \displaystyle \int_0^{\pi} \cos^{4}{\left(x \right)}\, dx \).
- Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, then integrate.Reviewed
- \[ \frac{d}{d x} \left(\frac{3 x}{8} + \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32}\right) = \cos^{4}{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
- \[ \frac{3 \pi}{8} \]Evaluate from 0.✓ Proved
Answer \( \frac{3 \pi}{8} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 |
| 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 | numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the method (power-reduction formulas) and verifies the result via differentiation and evaluation. The steps are logically sound and mathematically correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the method (power-reduction formulas) and verifies the result via differentiation and evaluation. The steps are logically sound and mathematically correct.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the method (power-reduction formulas) and the final result is verified by differentiation and evaluation.
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.