Integral of \( \displaystyle \cos{\left(2 x + 2 \right)} \)
Problem 4.882 · 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 simplifyStart with the integral of the function. Factor out the 2 inside the argument. The expression is equivalent to the original integrand.✓ Proved
- \[ = \frac{\sin{\left(2 x + 2 \right)}}{2} \]antiderivativeIntegrate the cosine function using the chain rule in reverse.✓ Proved
Answer \( \frac{\sin{\left(2 x + 2 \right)}}{2} + C \)
✓ Nihil obstat 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. 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 | ✓ 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: passqwen3.6:27b-mlx: fail (style) — Steps 2 and 3 are redundant and do not change the expression meaningfully; step 2 factors and step 3 immediately undoes it, violating the principle that each step should change one thing productively. Additionally, the label 'antiderivative' in step 4 is acceptable, but the note mentions 'chain rule in reverse' which is not a rule in the fixed vocabulary, though the label itself is valid.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-11 — Steps 2 and 3 are redundant and do not change the expression meaningfully; step 2 factors and step 3 immediately undoes it, violating the principle that each step should change one thing productively. Additionally, the label 'antiderivative' in step 4 is acceptable, but the note mentions 'chain rule in reverse' which is not a rule in the fixed vocabulary, though the label itself is valid.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (style) 2026-10-11 — Steps 2 and 3 are redundant and do not follow the 'one change' principle effectively; Step 2 factors the argument, and Step 3 immediately undoes it, adding no value. Furthermore, Step 4 applies the antiderivative rule but the note incorrectly attributes this to the 'chain rule in reverse' without explicitly showing the substitution step required by the contract's granularity rules (though 'antiderivative' is a valid label, the jump from step 3 to 4 skips the explicit substitution u = 2x+2 which is usually expected for non-trivial inner functions, or at least the label 'substitution' should be used if the change of variable is implicit). However, the primary defect is the redundant back-and-forth in steps 2 and 3.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.