Integral of \( \displaystyle \cos{\left(2 x - 2 \right)} \)
Problem 4.367 · easy
Find \( \displaystyle \int \cos{\left(2 x - 2 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos{\left(2 x - 2 \right)}\, dx \]integral algebra substitutionStart with the integral of the given function. Factor out the 2 from the argument. Let u = 2x - 2, then du = 2dx, so dx = du/2.✓ Proved
- \[ = \frac{\sin{\left(2 x - 2 \right)}}{2} \]antiderivativeApply the basic integration rule for cosine.✓ Proved
Answer \( \frac{\sin{\left(2 x - 2 \right)}}{2} + C \)
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. A reviewer's objection to the explanation stands; see the receipt.
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 | simplify(a - b) reduced to 0 |
| 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 | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (error) — Step 3 claims to perform a substitution but the integral is left unchanged; the factor 1/2 from du=2dx is not applied, leading to an incorrect antiderivative in step 4.qwen3.6:27b-mlx: fail (error) — Step 3 claims to perform a substitution but fails to transform the integrand or differential, leaving the expression identical to Step 1. Step 4 then jumps to the final answer without showing the intermediate integral in terms of u, violating the single-rule-per-step constraint and logical continuity.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 claims to perform a substitution but fails to transform the integrand or differential, leaving the expression identical to Step 1. Step 4 then jumps to the final answer without showing the intermediate integral in terms of u, violating the single-rule-per-step constraint and logical continuity.gpt-oss:20b: fail (error) 2026-10-05 — Step 3 claims to perform a substitution but the integral is left unchanged; the factor 1/2 from du=2dx is not applied, leading to an incorrect antiderivative in step 4.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 3 is labeled 'substitution' but fails to actually perform the substitution; it merely restates the integral from Step 1 without changing the variable of integration or the integrand. A valid substitution step must transform the integral into terms of the new variable (e.g., Integral(cos(u)/2, u)).gpt-oss:20b: inconclusive 2026-10-05 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 3 incorrectly applies the substitution rule: after setting \(u=2x-2\) the integral should become \(\frac12\int\cos(u)\,du\), not \(\int\cos(2x-2)\,dx
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.