Home›Calculus 2›Integrals of powers of sine and cosine›Problem 4.710

Integrals of powers of sine and cosine

Problem 4.710 · medium

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

The power of cosine is odd: keep one cos x for du, write the rest as (1 − sin²x)^1, and let u = sin x.
Reviewed
\[ \frac{d}{d u} \left(- \frac{u^{6}}{6} + \frac{u^{4}}{4}\right) = - u^{5} + u^{3} \]
∫ -u**5 + u**3 du = -u**6/6 + u**4/4.✓ Proved
\[ \frac{d}{d x} \left(- \frac{\sin^{6}{\left(x \right)}}{6} + \frac{\sin^{4}{\left(x \right)}}{4}\right) = \sin^{3}{\left(x \right)} \cos^{3}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( - \frac{\sin^{6}{\left(x \right)}}{6} + \frac{\sin^{4}{\left(x \right)}}{4} + C \)

✓ 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 odd powers of sine and cosine. The algebraic steps and final verification are correct.

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 odd powers of sine and cosine. The algebraic steps and final verification are correct.
gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the strategy for integrating odd powers of sine and cosine. The algebraic steps and final verification 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.