Integral of \( \displaystyle \left(2 x - 1\right)^{2} e^{\left(2 x - 1\right)^{3}} \)
Problem 4.1122 · medium Video
Find \( \displaystyle \int \left(2 x - 1\right)^{2} e^{\left(2 x - 1\right)^{3}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x - 1\right)^{2} e^{\left(2 x - 1\right)^{3}}\, dx \]integral algebra substitutionStart with the integral of the given function. Rearrange the integrand. Rewrite the integrand to include the derivative of the exponent.✓ Proved
- \[ = \frac{e^{\left(2 x - 1\right)^{3}}}{6} \]antiderivativeIntegrate the expression.✓ Proved
Answer \( \frac{e^{\left(2 x - 1\right)^{3}}}{6} + C \)
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 |
| 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 applies two rules at once (algebraic manipulation and substitution) but is labeled only as "substitution". Each step should change only one thing and label the rule correctly.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-oss:20b: fail (style) 2026-10-17 — Step 3 applies two rules at once (algebraic manipulation and substitution) but is labeled only as "substitution". Each step should change only one thing and label the rule correctly.qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 3 applies two rules simultaneously: it performs algebraic manipulation (multiplying by 6/6) and sets up a substitution, violating the one-rule-per-step constraint. Furthermore, Step 4 jumps from the prepared integral to the final answer without showing the substitution variable change or the integration step, effectively hiding the logic.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the substitution structure and applies the rules in a logical sequence. Step 3 prepares the integrand for recognition as a derivative, and Step 4 correctly identifies the antiderivative.gpt-oss:20b: fail (style) 2026-10-14 — Step 3 applies only an algebraic manipulation (multiplying and dividing by 6) but is labeled "substitution". The correct label for that step should be "algebra".
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 2026-10-17 with SymPy 1.14.0.