Integral of \( \displaystyle x \cos{\left(x^{2} \right)} \)
Problem 4.188 · easy Video
Find \( \displaystyle \int x \cos{\left(x^{2} \right)} \, dx \). (Omit the constant of integration.)
- \[ \int x \cos{\left(x^{2} \right)}\, dx \]integral algebra algebraStart with the integral of the given function. Multiply and divide by 2*x to prepare for substitution. Rearrange the terms for clarity.✓ Proved
- \[ = \frac{\int 2 x \cos{\left(x^{2} \right)}\, dx}{2} \]linearityPull the constant factor outside the integral.✓ Proved
- \[ = \frac{\sin{\left(x^{2} \right)}}{2} \]antiderivativeIntegrate using the substitution u = x**2, du = 2*x dx.✓ Proved
Answer \( \frac{\sin{\left(x^{2} \right)}}{2} + C \)
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. 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 |
| 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 (error) — Step 5 applies the antiderivative rule but fails to account for the factor of 2 in the differential (du = 2x dx), resulting in a missing factor of 1/2 in the final answer. The correct antiderivative is sin(x**2)/4.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the antiderivative rule but fails to account for the factor of 2 in the differential (du = 2x dx), resulting in a missing factor of 1/2 in the final answer. The correct antiderivative is sin(x**2)/4.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution is correct and follows the contract. Step 2 introduces a factor of 1 via multiplication by 2x/(2x), which is a valid algebraic manipulation. Step 5 correctly identifies the antiderivative using the substitution implied by the previous steps, and the label 'antiderivative' is appropriate for recognizing the integral of a composite function with its derivative present.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 5 applies both the substitution rule and the antiderivative rule simultaneously, violating the one-rule-per-step constraint. It should be split into a substitution step and an antiderivative step.gpt-oss:20b: pass 2026-09-29
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 2026-10-03 with SymPy 1.14.0.