Integral of \( \displaystyle \cos{\left(6 x + 2 \right)} \)
Problem 4.479 · easy
Find \( \displaystyle \int \cos{\left(6 x + 2 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos{\left(6 x + 2 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = 6 \int \frac{\cos{\left(6 x + 2 \right)}}{6}\, dx \]algebra linearityRewrite the integrand by multiplying and dividing by 6. Move the constant factor outside the integral.✓ Proved
- \[ = \int \cos{\left(6 x + 2 \right)}\, dx \]algebra simplifyRearrange the terms. Simplify the constant factor.✓ Proved
- \[ = \frac{\sin{\left(6 x + 2 \right)}}{6} \]antiderivativeCompute the antiderivative.✓ Proved
Answer \( \frac{\sin{\left(6 x + 2 \right)}}{6} + C \)
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. 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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ 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 2 applies two transformations at once: it rewrites the integrand by multiplying/dividing by 6 and simultaneously multiplies the entire integral by 6. Each change must be a separate step according to the contract.qwen3.6:27b-mlx: fail (error) — Step 2 is mathematically invalid: it claims Integral(f) = Integral(f * 1/6) * 6, which implies Integral(f) = 1/6 * Integral(f) * 6, but the placement of the factor 6 outside the integral in the expression 'Integral(...) * 6' is ambiguous and structurally incorrect for standard integral notation without parentheses. More critically, Step 4 rearranges '6 * Integral(cos(6*x + 2) * (1/6), x)' into '6 * Integral(cos(6*x + 2), x) * (1/6)', which incorrectly moves the constant 1/6 outside the integral sign without adjusting the integral operator's scope properly, or implies that Integral(f * c) = Integral(f) * c inside a product with another constant, which is not a standard algebraic manipulation of the integral symbol itself. The standard substitution method would introduce a new variable u = 6x+2. The current steps mix algebraic manipulation of the integrand with improper handling of the integral operator.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — Step 2 applies two transformations at once: it rewrites the integrand by multiplying/dividing by 6 and simultaneously multiplies the entire integral by 6. Each change must be a separate step according to the contract.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 2 is mathematically invalid: it claims Integral(f) = Integral(f * 1/6) * 6, which implies Integral(f) = 1/6 * Integral(f) * 6, but the placement of the factor 6 outside the integral in the expression 'Integral(...) * 6' is ambiguous and structurally incorrect for standard integral notation without parentheses. More critically, Step 4 rearranges '6 * Integral(cos(6*x + 2) * (1/6), x)' into '6 * Integral(cos(6*x + 2), x) * (1/6)', which incorrectly moves the constant 1/6 outside the integral sign without adjusting the integral operator's scope properly, or implies that Integral(f * c) = Integral(f) * c inside a product with another constant, which is not a standard algebraic manipulation of the integral symbol itself. The standard substitution method would introduce a new variable u = 6x+2. The current steps mix algebraic manipulation of the integrand with improper handling of the integral operator.gpt-oss:20b: fail (error) 2026-10-06 — Step 2 applies two transformations at once: it rewrites the integrand by multiplying by 1/6 and simultaneously multiplies the entire integral by 6. Each step must change only one thing, so this is a defect.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 2 is mathematically incorrect: Integral(f) * 6 is not equal to Integral(f * 1/6) * 6 (the latter is Integral(f)/6). The solver confused the constant multiple rule, effectively dividing by 36 instead of 6, or simply wrote an invalid equality.
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 2026-10-06 with SymPy 1.14.0.