Integrals of powers of sine and cosine
Problem 4.706 · medium
Evaluate \( \displaystyle \int_0^{\pi} \sin^{3}{\left(x \right)}\, dx \).
- The power of sine is odd: keep one sin x for du, write the rest as (1 − cos²x)^1, and let u = cos x (du = −sin x dx).
- \[ \frac{d}{d u} \left(\frac{u^{3}}{3} - u\right) = u^{2} - 1 \]∫ u**2 - 1 du = u**3/3 - u.✓ Proved
- \[ \frac{d}{d x} \left(\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}\right) = \sin^{3}{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
- \[ \frac{4}{3} \]Evaluate from 0.✓ Proved
Answer \( \frac{4}{3} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly perform the definite integration steps (substituting limits into the antiderivative). While the final numerical answer is correct, the logical gap between finding the indefinite integral and stating the final value makes the solution incomplete and potentially misleading regarding the evaluation process.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly perform the definite integration steps (substituting limits into the antiderivative). While the final numerical answer is correct, the logical gap between finding the indefinite integral and stating the final value makes the solution incomplete and potentially misleading regarding the evaluation process.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the standard technique for integrating odd powers of sine. The algebraic steps and final evaluation are correct.
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 2026-10-08 with SymPy 1.14.0.