Home›Calculus 2›Integrals of powers of sine and cosine›Problem 4.569

Integrals of powers of sine and cosine

Problem 4.569 · easy

Evaluate \( \displaystyle \int_0^{\frac{\pi}{2}} \sin^{2}{\left(x \right)}\, dx \).

Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, then integrate.
Reviewed
\[ \frac{d}{d x} \left(\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}\right) = \sin^{2}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved
\[ \frac{\pi}{4} \]
Evaluate from 0.✓ Proved

Answer \( \frac{\pi}{4} \)

Lines: 2 proved, 1 reviewed. 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	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	numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the power-reduction identity for sin²(x) and verifies the antiderivative and evaluation. The mention of cos²x in the first sentence is irrelevant but not incorrect.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the power-reduction identity for sin²(x) and verifies the antiderivative and evaluation. The mention of cos²x in the first sentence is irrelevant but not incorrect.
gpt-oss:20b: fail (misleading) 2026-10-07 — The first sentence incorrectly references cos²x and states "Both powers are even", which is irrelevant and may confuse students. The correct approach is to use sin²x = (1−cos2x)/2 only.
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the power-reduction identity for sin²(x) and verifies the antiderivative and evaluation. The mention of cos²x in the first sentence is unnecessary but not incorrect.

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 2026-10-07 with SymPy 1.14.0.