Integral of \( \displaystyle \left(2 x - 2\right) \sqrt{\left(x - 1\right)^{2} + 1} \)
Problem 4.919 · medium
Find \( \displaystyle \int 2 \left(x - 1\right) \sqrt{\left(x - 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x - 2\right) \sqrt{\left(x - 1\right)^{2} + 1}\, dx \]integral rewriteStart with the integral of the given function. Rewrite the square root using exponent notation.✓ Proved
- \[ = \frac{2 \left(\left(x - 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} \]antiderivativeApply the power rule for integration.✓ Proved
Answer \( \frac{2 \left(x^{2} - 2 x + 2\right)^{\frac{3}{2}}}{3} + C \)
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 |
| 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 (style) — Step 3 combines a substitution (u=(x-1)^2+1) with the power rule in a single line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) — Step 3 applies the power rule for integration directly to the composite function without accounting for the chain rule (the derivative of the inner function is 2(x-1), which matches the outer factor, but the step skips the substitution logic required to justify the antiderivative). The label 'antiderivative' is insufficient because the step performs a non-trivial integration of a composite function rather than just recalling a basic form, and it fails to show the necessary substitution or chain-rule reversal explicitly as a single logical step.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-12 — Step 3 combines a substitution (u=(x-1)^2+1) with the power rule in a single line, violating the rule that each step must change only one thing.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 3 applies the power rule for integration directly to the composite function without accounting for the chain rule (the derivative of the inner function is 2(x-1), which matches the outer factor, but the step skips the substitution logic required to justify the antiderivative). The label 'antiderivative' is insufficient because the step performs a non-trivial integration of a composite function rather than just recalling a basic form, and it fails to show the necessary substitution or chain-rule reversal explicitly as a single logical step.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 3 applies the antiderivative rule incorrectly by ignoring the chain rule factor (the derivative of the inner function). The step jumps from the integrand directly to the result without accounting for the missing factor of 2 from the inner derivative, or equivalently, it fails to show the substitution step required to justify the power rule application.gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.