Substitution with a given u
Problem 4.316 · hard
Evaluate \( \displaystyle \int 3 \sin^{6}{\left(2 x \right)} \cos{\left(2 x \right)}\, dx \) 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 ∫ 3*u**6/2 du.Reviewed
- \[ \frac{d}{d u} \frac{3 u^{7}}{14} = \frac{3 u^{6}}{2} \]An antiderivative in u is 3*u**7/14; differentiating confirms it.✓ Proved
- \[ \frac{3 \sin^{7}{\left(2 x \right)}}{14} \]Replace u by sin(2*x).✓ Proved
- \[ \frac{d}{d x} \frac{3 \sin^{7}{\left(2 x \right)}}{14} = 3 \sin^{6}{\left(2 x \right)} \cos{\left(2 x \right)} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{3 \sin^{7}{\left(2 x \right)}}{14} + C \)
✓ 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, handling the constant factor from the differential accurately. The final result matches the stated answer and is verified by differentiation.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the substitution method, handling the constant factor from the differential accurately. The final result matches the stated answer and is verified by differentiation.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the substitution method, handles the constant factor from the differential, and verifies the result.gpt-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.