Integrals of powers of sine and cosine
Problem 4.1021 · medium
Evaluate \( \displaystyle \int_0^{\frac{\pi}{4}} \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx \).
- The power of sine is odd: keep one sin x for du, write the rest as (1 − cos²x)^2, and let u = cos x (du = −sin x dx).Reviewed
- \[ \frac{d}{d u} \left(- \frac{u^{7}}{7} + \frac{2 u^{5}}{5} - \frac{u^{3}}{3}\right) = - u^{6} + 2 u^{4} - u^{2} \]∫ -u**6 + 2*u**4 - u**2 du = -u**7/7 + 2*u**5/5 - u**3/3.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{\cos^{7}{\left(x \right)}}{7} + \frac{2 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}\right) = \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
- \[ - \frac{71 \sqrt{2}}{1680} + \frac{8}{105} = \frac{8}{105} - \frac{71 \sqrt{2}}{1680} \]Evaluate from 0.✓ Proved
Answer \( \frac{8}{105} - \frac{71 \sqrt{2}}{1680} \)
Lines: 3 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 |
| 4 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the substitution method for odd powers of sine, performs the integration accurately, and evaluates the definite integral to match the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the substitution method for odd powers of sine, performs the integration accurately, and evaluates the definite integral to match the stated answer.gpt-oss:20b: fail (misleading) 2026-10-13 — The solution omits the crucial step of evaluating the antiderivative at the limits 0 and π/4; it merely repeats the final expression without showing the computation, which could mislead a student into thinking the integral has been fully evaluated.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies the standard substitution method for integrals of odd powers of sine. The algebraic manipulation, integration, and final evaluation 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 2026-10-13 with SymPy 1.14.0.