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Integrals of powers of sine and cosine

Problem 4.709 · medium

Evaluate \( \displaystyle \int \sin^{3}{\left(x \right)}\, dx \).

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).
Reviewed
\[ \frac{d}{d u} \left(\frac{u^{3}}{3} - u\right) = u^{2} - 1 \]
∫ 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

Answer \( \frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)} + C \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the strategy for integrating an odd power of sine, performs the substitution and integration accurately, and verifies the result by differentiation.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the strategy for integrating an odd power of sine, performs the substitution and integration accurately, and verifies the result by differentiation.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the standard technique for integrating odd powers of sine. The substitution steps and final verification are accurate.

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.