Integral of \( \displaystyle 2 x \cos{\left(4 x^{2} \right)} \)
Problem 4.1243 · medium Video
Find \( \displaystyle \int 2 x \cos{\left(4 x^{2} \right)} \, dx \). (Omit the constant of integration.)
- \[ \int 2 x \cos{\left(4 x^{2} \right)}\, dx \]integral rewrite algebra linearityStart with the integral of the given function. Rewrite the integrand to highlight the inner function's derivative. Express 2*x as (4*x)/2. Pull the constant factor out of the integral.✓ Proved
- \[ = \frac{\int 4 x \cos{\left(4 x^{2} \right)}\, dx}{2} \]linearityApply linearity to the constant factor.✓ Proved
- \[ = \frac{\sin{\left(4 x^{2} \right)}}{4} \]antiderivative simplifyThe antiderivative of 4*x*cos(4*x**2) is sin(4*x**2)/2. Simplify the final expression.✓ Proved
Answer \( \frac{\sin{\left(4 x^{2} \right)}}{4} + C \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
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: passqwen3.6:27b-mlx: pass — The solution correctly applies substitution via algebraic manipulation and linearity, with each step changing only one aspect of the expression and using valid labels.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies substitution via algebraic manipulation and linearity, with each step changing only one aspect of the expression and using valid labels.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies substitution logic via algebraic manipulation and linearity, with each step changing only one aspect of the expression and using valid labels.gpt-oss:20b: fail (error) 2026-10-16 — Step 6 applies two rules at once: it both evaluates the antiderivative of 4*x*cos(4*x**2) and multiplies by the preceding 1/2. The label "antiderivative" does not account for the multiplication, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies substitution via algebraic manipulation and linearity. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.