Integral of \( \displaystyle \cos{\left(2 x \right)} \)
Problem 4.267 · easy
Find \( \displaystyle \int \cos{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos{\left(2 x \right)}\, dx \]integral algebra algebraStart with the integral of the function. Prepare for substitution by multiplying by 1. Introduce the constant 2 to facilitate substitution.✓ Proved
- \[ = \frac{\int 2 \cos{\left(2 x \right)}\, dx}{2} \]linearityMove the constant outside the integral.✓ Proved
- \[ = \int \cos{\left(2 x \right)}\, dx \]algebra simplifySimplify the expression inside the integral. The constants cancel out.✓ Proved
- \[ = \frac{\sin{\left(2 x \right)}}{2} \]antiderivativeCompute the antiderivative of cos(2x).✓ Proved
Answer \( \frac{\sin{\left(2 x \right)}}{2} + C \)
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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ 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 5 applies two transformations at once: it pulls the factor 2 out of the integral and then multiplies the result by 2, violating the rule‑granularity requirement. Additionally, step 6 incorrectly claims the constants cancel, leaving the integral unchanged without justification.qwen3.6:27b-mlx: fail (error) — Steps 5 and 6 reverse the progress made in steps 3 and 4, returning to the original integral without performing the substitution. Step 7 then claims to compute the antiderivative of cos(2x) directly, but the label 'antiderivative' is reserved for unfolding d/dx on a known form (like sin(x) or e^x), not for handling the chain rule coefficient 2. The solution fails to actually perform the substitution u=2x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Steps 5 and 6 reverse the progress made in steps 3 and 4, returning to the original integral without performing the substitution. Step 7 then claims to compute the antiderivative of cos(2x) directly, but the label 'antiderivative' is reserved for unfolding d/dx on a known form (like sin(x) or e^x), not for handling the chain rule coefficient 2. The solution fails to actually perform the substitution u=2x.gpt-oss:20b: fail (error) 2026-10-04 — Step 5 applies two transformations at once: it pulls the factor 2 out of the integral and then multiplies the result by 2, violating the rule‑granularity requirement. Additionally, step 6 incorrectly claims the constants cancel, leaving the integral unchanged without justification.qwen3.6:27b-mlx: fail (error) 2026-10-04 — Steps 5 and 6 are mathematically invalid; one cannot arbitrarily move the factor 2 outside the integral or cancel constants in that manner without changing the value of the expression. The solution fails to actually perform the substitution or recognize the derivative of the inner function.gpt-oss:20b: fail (error) 2026-10-04 — Step 5 applies two operations at once: it pulls the factor 2 out of the integral and then multiplies it back in, which is not a single allowed rule. The subsequent cancellation in step 6 is therefore unjustified.
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 2026-10-04 with SymPy 1.14.0.