Integrals of powers of sine and cosine
Problem 4.862 · medium
Evaluate \( \displaystyle \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx \).
- The power of cosine is odd: keep one cos x for du, write the rest as (1 − sin²x)^0, and let u = sin x.Reviewed
- \[ \frac{d}{d u} \frac{u^{5}}{5} = u^{4} \]∫ 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
Answer \( \frac{\sin^{5}{\left(x \right)}}{5} + C \)
✓ 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 | difference quotients of the answer match the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the substitution method for an odd power of cosine and verifies the result by differentiation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the substitution method for an odd power of cosine and verifies the result by differentiation.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the substitution method for an odd power of cosine and verifies the result by 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.