Integral of \( \displaystyle \left(3 x - 1\right) e^{\left(3 x - 1\right)^{2}} \)
Problem 4.1339 · medium Video
Find \( \displaystyle \int \left(3 x - 1\right) e^{\left(3 x - 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(3 x - 1\right) e^{\left(3 x - 1\right)^{2}}\, dx \]integral algebra algebra algebra algebra algebra simplify algebraStart with the integral of the given function. Multiply and divide by 6 to create the derivative of the exponent in the integrand. Distribute the 6 into the polynomial term. Factor out a 2. Factor out a 3. Simplify the constant product. Simplify the integrand back to its original form to verify the substitution. Rewrite to prepare for the substitution u = (3x-1)**2.✓ Proved
- \[ = \frac{e^{\left(3 x - 1\right)^{2}}}{6} \]antiderivativeApply the antiderivative rule for the exponential function.✓ Proved
Answer \( \frac{e^{\left(3 x - 1\right)^{2}}}{6} + C \)
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. A reviewer's objection to the explanation stands; see 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 |
| 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 8 introduces an unnecessary factor of 2/2 and Step 9 incorrectly applies the antiderivative rule for exp((3*x-1)**2). The correct antiderivative is exp((3*x-1)**2)/6, but the steps do not justify this factor correctly, violating the rule‑application granularity and producing a mathematically incorrect intermediate step.qwen3.6:27b-mlx: fail (error) — Step 9 jumps from the integral to the final answer without performing the substitution or integration steps, and the label 'antiderivative' is not in the allowed vocabulary (which requires 'substitution' or 'integral' for such operations). Additionally, steps 2-7 are circular and do not progress toward the solution.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-25 — Step 9 jumps from the integral to the final answer without performing the substitution or integration steps, and the label 'antiderivative' is not in the allowed vocabulary (which requires 'substitution' or 'integral' for such operations). Additionally, steps 2-7 are circular and do not progress toward the solution.gpt-oss:20b: fail (error) 2026-10-25 — Step 8 introduces an unnecessary factor of 2/2 and Step 9 incorrectly applies the antiderivative rule for exp((3*x-1)**2). The correct antiderivative is exp((3*x-1)**2)/6, but the steps do not justify this factor correctly, violating the rule‑application granularity and producing a mathematically incorrect intermediate step.qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 9 is mathematically incorrect; the antiderivative of exp((3x-1)^2) * (3x-1) is exp((3x-1)^2)/6, but the step claims the result is 1/6 * exp((3x-1)^2) without accounting for the chain rule factor of 6 in the denominator relative to the inner derivative, or rather, it applies the 'antiderivative' rule incorrectly by ignoring the necessary division by the derivative of the inner function's square (which is 2*(3x-1)*3 = 18*(3x-1), wait, let's re-evaluate). Actually, let's look closer. The integrand is (3x-1)e^{(3x-1)^2}. Let u = (3x-1)^2, du = 2(3x-1)*3 dx = 6(3x-1)dx. So (3x-1)dx = du/6. The integral is int e^u du/6 = e^u/6 = e^{(3x-1)^2}/6. Step 9 says: Integral(exp((3*x - 1)**2) * (3*x - 1) * (2/2), x) -> 1/6 * exp((3*x - 1)**2). Wait, Step 8 introduced a factor of 2/2. So the integrand is 2/2 * (3x-1) * exp(...). If we use substitution u=(3x-1)^2, du = 6(3x-1)dx. Then (3x-1)dx = du/6. The integral is int 2/2 * e^u * (du/6) = 1/6 int e^u du = 1/6 e^u. So the result 1/6 exp((3x-1)^2) is actually correct. However, the step label is 'antiderivative'. The step jumps from the integral expression to the final answer. It combines substitution and integration. The contract says 'Each step changes ONE thing'. Step 9 performs substitution (implicitly) AND integration. This violates the one-rule-per-step constraint. Also, the note says 'Apply the antiderivative rule for the exponential function', which is misleading because it ignores the chain rule/substitution mechanics entirely. A student would not know where the 1/6 came from if they just followed the note. Let's check the previous steps. Steps 2-7 are a bizarre loop that multiplies by 6/6, expands, factors, simplifies, and returns to the start. This is logically valid but stylistically terrible. Step 8 multiplies by 2/2. Step 9 jumps to the answer. The primary defect is Step 9. It applies substitution and integration in one step. The label 'antiderivative' suggests a direct lookup, but the coefficient 1/6 requires recognizing the derivative of the exponent. The step does not show the substitution variable change. It is a 'magic' step. Is it an error or style? The prompt says 'A step that applies two rules at once is a defect.' Step 9 applies substitution (algebra/rewrite) and integration (antiderivative). It is a defect. Also, Step 7 is redundant and confusing, but not a defect per se. The verdict is fail due to Step 9 applying multiple rules (substitution and integration) at once, violating the one-rule-per-step contract.gpt-oss:20b: inconclusive 2026-10-24 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution incorrectly applies the antiderivative rule in step 9. The integrand is \((3x-1)\exp((3x-1)^2)\), whose derivative of the exponent is \(6(3x-qwen3.6:27b-mlx: fail (error) 2026-10-24 — The solution contains a logical loop where steps 3-7 perform unnecessary algebraic manipulations that return to the original integrand, failing to actually execute the substitution. Step 9 jumps directly to the final answer without showing the substitution step (e.g., letting u = (3x-1)^2) or applying the chain rule in reverse, making the derivation incomplete and logically disconnected.gpt-oss:20b: fail (error) 2026-10-24 — Step 8 incorrectly rewrites the integrand as (3*x-1)*exp((3*x-1)**2)*(2/2). This does not produce the factor 1/6 needed to match u' = 6*(3*x-1). Consequently, step 9 applies the antiderivative rule to an integrand that is not of the form (1/6)*u'*exp(u), leading to an incorrect antiderivative.
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 2026-10-25 with SymPy 1.14.0.