Home›Calculus 2›Integrals of powers of sine and cosine›Problem 4.572

Integrals of powers of sine and cosine

Problem 4.572 · medium

Evaluate \( \displaystyle \int_0^{\frac{\pi}{4}} \sin^{4}{\left(x \right)}\, dx \).

Both powers are even: use sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2, then integrate.
Reviewed
\[ \frac{d}{d x} \left(\frac{3 x}{8} - \frac{\sin{\left(2 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{32}\right) = \sin^{4}{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved
\[ - \frac{1}{4} + \frac{3 \pi}{32} \]
Evaluate from 0.✓ Proved

Answer \( - \frac{1}{4} + \frac{3 \pi}{32} \)

✓ 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	numerical quadrature

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the method for reducing even powers of sine and verifies the result via differentiation and evaluation. The steps are logically sound and mathematically correct.

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 method for reducing even powers of sine and verifies the result via differentiation and evaluation. The steps are logically sound and mathematically correct.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the method (power reduction) and verifies the result via differentiation and evaluation. The sentence accurately describes the standard approach for even powers of sine.

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.