Integral of \( \displaystyle \sin{\left(2 x - 1 \right)} \cos{\left(2 x - 1 \right)} \)
Problem 4.374 · medium
Find \( \displaystyle \int \sin{\left(2 x - 1 \right)} \cos{\left(2 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(2 x - 1 \right)} \cos{\left(2 x - 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{\sin{\left(4 x - 2 \right)}}{2}\, dx \]trig-identity algebraUse the double angle identity sin(2u) = 2sin(u)cos(u). Distribute the 2 inside the sine function.≈ Checked numerically
- \[ = - \frac{\cos{\left(4 x - 2 \right)}}{8} \]antiderivativeIntegrate the sine function.✓ Proved
Answer \( \frac{\sin^{2}{\left(2 x - 1 \right)}}{4} + C \)
✓ Nihil obstat Lines: 3 proved, 2 checked numerically. 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 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left sin(2*x - 1)**2/4 + cos(4*x - 2)/8; numeric agreement only, at 24 of 24 sampled points |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -sin(2*x - 1)**2/4 - cos(4*x - 2)/8; numeric agreement only, at 24 of 24 sampled points |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the double-angle identity, simplifies the argument, and integrates. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the double-angle identity, simplifies the argument, and integrates. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the double-angle identity, simplifies the argument, and integrates. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-05
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
gemma4:26b, checked 2026-10-05 with SymPy 1.14.0.