Integral of \( \displaystyle \sin{\left(3 x - 1 \right)} \cos{\left(3 x - 1 \right)} \)
Problem 4.1242 · medium Video
Find \( \displaystyle \int \sin{\left(3 x - 1 \right)} \cos{\left(3 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(3 x - 1 \right)} \cos{\left(3 x - 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{\sin{\left(6 x - 2 \right)}}{2}\, dx \]trig-identity algebraUse the double angle identity sin(2u) = 2sin(u)cos(u). Distribute the 2 inside the sine function.✓ Proved
- \[ = \frac{\int \sin{\left(6 x - 2 \right)}\, dx}{2} \]linearityFactor out the constant 1/2.✓ Proved
- \[ = - \frac{\cos{\left(6 x - 2 \right)}}{12} \]antiderivativeIntegrate the sine function using the substitution rule for linear arguments.✓ Proved
Answer \( - \frac{\cos^{2}{\left(3 x - 1 \right)}}{6} + C \)
Every line of this solution was proved by the computer algebra system SymPy. 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -1/12 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: lines differ by the constant 1/12 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (error) — The final antiderivative is missing the constant term that arises when rewriting -cos(3*x-1)**2/6 in terms of sin(6*x-2). The computed result -cos(6*x-2)/12 is not equivalent to the stated answer up to an additive constant, so the solution is mathematically incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19gpt-oss:20b: fail (error) 2026-10-19 — The final antiderivative is missing the constant term that arises when rewriting -cos(3*x-1)**2/6 in terms of sin(6*x-2). The computed result -cos(6*x-2)/12 is not equivalent to the stated answer up to an additive constant, so the solution is mathematically incorrect.qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.