Integral of \( \displaystyle \left(x - 1\right) \cos{\left(\left(x - 1\right)^{2} \right)} \)
Problem 4.1001 · medium
Find \( \displaystyle \int \left(x - 1\right) \cos{\left(\left(x - 1\right)^{2} \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(x - 1\right) \cos{\left(\left(x - 1\right)^{2} \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(x - 1\right) \cos{\left(x^{2} - 2 x + 1 \right)}\, dx \]algebraExpand the squared term inside the cosine.✓ Proved
- \[ = \int \left(x \cos{\left(x^{2} - 2 x + 1 \right)} - \cos{\left(x^{2} - 2 x + 1 \right)}\right)\, dx \]linearityDistribute the term (x-1) across the cosine.✓ Proved
- \[ = \int \left(x \cos{\left(\left(x - 1\right)^{2} \right)} - \cos{\left(\left(x - 1\right)^{2} \right)}\right)\, dx \]algebraRewrite the polynomial in terms of the squared binomial.✓ Proved
- \[ = \int x \cos{\left(\left(x - 1\right)^{2} \right)}\, dx - \int \cos{\left(\left(x - 1\right)^{2} \right)}\, dx \]linearity algebraSplit the integral into two parts. Express x as (x - 1) + 1 to facilitate substitution.✓ Proved
- \[ = \int \left(x - 1\right) \cos{\left(\left(x - 1\right)^{2} \right)}\, dx \]linearity simplifyDistribute the first integral over the sum. The extra integral terms cancel out.✓ Proved
- \[ = 0.5 \sin{\left(\left(x - 1\right)^{2} \right)} \]substitutionUse substitution u = (x - 1)**2, then du = 2(x - 1)dx.✓ Proved
Answer \( \frac{\sin{\left(\left(x - 1\right)^{2} \right)}}{2} + C \)
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.
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 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 7 incorrectly distributes the integral over a sum that contains an integral itself, resulting in an expression with nested integrals. This violates the rule that each step may only apply one allowed rule to one term, and it introduces a mathematically invalid operation.qwen3.6:27b-mlx: fail (error) — Step 7 is mathematically invalid; it incorrectly distributes the integral operator over a sum inside the integrand while simultaneously adding and subtracting an integral term outside, violating the single-rule constraint and producing a nonsensical expression. Step 9 then claims to solve the original integral via substitution, but the previous steps created a circular loop (Step 8 returns to the start) rather than solving it.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — Step 7 incorrectly distributes the integral over a sum that contains an integral itself, resulting in an expression with nested integrals. This violates the rule that each step may only apply one allowed rule to one term, and it introduces a mathematically invalid operation.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 7 is mathematically invalid; it incorrectly distributes the integral operator over a sum inside the integrand while simultaneously adding and subtracting an integral term outside, violating the single-rule constraint and producing a nonsensical expression. Step 9 then claims to solve the original integral via substitution, but the previous steps created a circular loop (Step 8 returns to the start) rather than solving it.gpt-oss:20b: fail (style) 2026-10-13 — Step 3 incorrectly labels the distributive expansion of (x-1)*cos(...) as "linearity"; the correct label is "algebra". This mislabeling could mislead a student about which rule is being applied.qwen3.6:27b-mlx: fail (error) 2026-10-13 — Step 7 is mathematically invalid; it incorrectly distributes the integral operator over a sum inside the integrand while simultaneously introducing and canceling terms in a way that violates the linearity rule (it treats the integral of a sum as the sum of integrals but applies it to a single integral term incorrectly, resulting in a nonsensical expression `Integral(A + B - B, x)` where the B terms are separate integrals, not part of the integrand). Step 8 then simplifies this invalid expression back to the original integral, creating a circular and logically broken argument.
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 2026-10-13 with SymPy 1.14.0.