Substitution with a given u
Problem 4.312 · hard
Evaluate \( \displaystyle \int 2 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \) using the substitution \( \displaystyle u = \sin{\left(2 x \right)} \).
- \[ \frac{d}{d x} \sin{\left(2 x \right)} = 2 \cos{\left(2 x \right)} \]So du = 2cos(2x) dx.✓ Proved
- In terms of u the integral becomes ∫ u**2 du.Reviewed
- \[ \frac{d}{d u} \frac{u^{3}}{3} = u^{2} \]An antiderivative in u is u**3/3; differentiating confirms it.✓ Proved
- \[ \frac{\sin^{3}{\left(2 x \right)}}{3} \]Replace u by sin(2*x).✓ Proved
- \[ \frac{d}{d x} \frac{\sin^{3}{\left(2 x \right)}}{3} = 2 \sin^{2}{\left(2 x \right)} \cos{\left(2 x \right)} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\sin^{3}{\left(2 x \right)}}{3} + C \)
✓ Nihil obstat Lines: 4 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | a difference quotient of the answer matches the integrand at x = 0.4, 0.9, 1.3 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the substitution method, handles the differential transformation accurately, and verifies the result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the substitution method, handles the differential transformation accurately, and verifies the result.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04
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/indefinite_substitution, checked 2026-10-04 with SymPy 1.14.0.