Integral of \( \displaystyle \cos{\left(4 x \right)} \)
Problem 4.650 · easy
Find \( \displaystyle \int \cos{\left(4 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos{\left(4 x \right)}\, dx \]integral algebra algebraStart with the integral of the function. Rewrite the integrand to prepare for substitution. Group the constant terms.✓ Proved
- \[ = 4 \int \frac{\cos{\left(4 x \right)}}{4}\, dx \]linearityPull the constant out of the integral.✓ Proved
- \[ = \frac{\sin{\left(4 x \right)}}{4} \]antiderivativeIntegrate the remaining expression.✓ Proved
Answer \( \frac{\sin{\left(4 x \right)}}{4} + C \)
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 |
| 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 the antiderivative rule incorrectly: the integral of (1/4)·cos(4x) is (1/4)·(1/4)·sin(4x)=1/16 sin(4x). Multiplying by the external factor 4 gives sin(4x), not 1/4 sin(4x).qwen3.6:27b-mlx: fail (error) — Step 4 is mathematically incorrect; pulling a constant factor out of an integral does not multiply the integral by the reciprocal of that constant (the factor 4 should not appear as a multiplier outside). Step 5 is also incorrect because it claims to integrate cos(4*x) to sin(4*x) without accounting for the chain rule factor.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — Step 5 applies the antiderivative rule incorrectly: the integral of (1/4)·cos(4x) is (1/4)·(1/4)·sin(4x)=1/16 sin(4x). Multiplying by the external factor 4 gives sin(4x), not 1/4 sin(4x).qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 4 is mathematically incorrect; pulling a constant factor out of an integral does not multiply the integral by the reciprocal of that constant (the factor 4 should not appear as a multiplier outside). Step 5 is also incorrect because it claims to integrate cos(4*x) to sin(4*x) without accounting for the chain rule factor.gpt-oss:20b: fail (error) 2026-10-08 — The final step drops the factor of 4 that was pulled out in step 4, yielding 1/4 sin(4x) instead of the correct sin(4x).qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 4 is mathematically incorrect: it claims to pull out a factor of 4, but the integrand in Step 3 contains a factor of 1/4. The correct application of linearity would yield (1/4) * Integral(4*cos(4*x), x), not 4 * Integral(cos(4*x)/4, x). The algebraic manipulation of constants is flawed.
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 2026-10-08 with SymPy 1.14.0.