Home›Calculus 2›Integrals of powers of sine and cosine›Problem 4.411

Integrals of powers of sine and cosine

Problem 4.411 · medium

Evaluate \( \displaystyle \int \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx \).

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
\[ \frac{d}{d u} \left(\frac{u^{5}}{5} - \frac{u^{3}}{3}\right) = u^{4} - u^{2} \]
∫ u**4 - u**2 du = u**5/5 - u**3/3.✓ Proved
\[ \frac{d}{d x} \left(\frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}\right) = \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3} + C \)

✓ 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 verifies the result via differentiation. The logic is sound and complete.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the strategy for integrating odd powers of sine and verifies the result via differentiation. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the strategy for integrating odd powers of sine and verifies the result via differentiation. The steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.