Integral of \( \displaystyle \left(2 x - 1\right)^{2} e^{2 x - 1} \)
Problem 4.152 · medium Video
Find \( \displaystyle \int \left(2 x - 1\right)^{2} e^{2 x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x - 1\right)^{2} e^{2 x - 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(4 x^{2} - 4 x + 1\right) e^{2 x - 1}\, dx \]algebraExpand the squared binomial.✓ Proved
- \[ = - \int 4 x e^{2 x - 1}\, dx + \int 4 x^{2} e^{2 x - 1}\, dx + \int e^{2 x - 1}\, dx \]linearityDistribute the exponential term and split the integral.✓ Proved
- \[ = \frac{e^{2 x - 1}}{2} - \int 4 x e^{2 x - 1}\, dx + \int 4 x^{2} e^{2 x - 1}\, dx \]antiderivativeIntegrate the constant term.✓ Proved
- \[ = - 2 x e^{2 x - 1} + \frac{e^{2 x - 1}}{2} + \int 4 x^{2} e^{2 x - 1}\, dx + \int 2 e^{2 x - 1}\, dx \]parts algebraApply integration by parts to the second term. Simplify the result of integration by parts.✓ Proved
- \[ = - 2 x e^{2 x - 1} + \frac{3 e^{2 x - 1}}{2} + \int 4 x^{2} e^{2 x - 1}\, dx \]antiderivative simplifyIntegrate the remaining integral. Combine the exponential terms.✓ Proved
- \[ = 2 x^{2} e^{2 x - 1} - 2 x e^{2 x - 1} + \frac{3 e^{2 x - 1}}{2} - \int 4 x e^{2 x - 1}\, dx \]partsApply integration by parts to the first term.✓ Proved
- \[ = 2 x^{2} e^{2 x - 1} - 4 x e^{2 x - 1} + \frac{3 e^{2 x - 1}}{2} + \int 2 e^{2 x - 1}\, dx \]partsApply integration by parts to the integral of x*exp(2x-1).✓ Proved
- \[ = 2 x^{2} e^{2 x - 1} - 4 x e^{2 x - 1} + \frac{5 e^{2 x - 1}}{2} \]antiderivative simplifyIntegrate the remaining exponential term. Combine all terms into the final antiderivative.✓ Proved
Answer \( \frac{\left(4 x^{2} - 8 x + 5\right) e^{2 x - 1}}{2} + C \)
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 |
| 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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 9 applies integration by parts to the first term but fails to account for the constant factor of 4, resulting in an incorrect coefficient for the remaining integral term. The step claims to apply 'parts' but the algebraic result is mathematically incorrect relative to the previous line.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 9 applies integration by parts to the first term but fails to account for the constant factor of 4, resulting in an incorrect coefficient for the remaining integral term. The step claims to apply 'parts' but the algebraic result is mathematically incorrect relative to the previous line.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 9 applies integration by parts to the first term but fails to include the factor of 4 from the integrand `4*x**2`, resulting in `2*x**2` instead of `4*x**2`. This is a mathematical error in the application of the rule.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 9 applies integration by parts to the first term but fails to include the factor of 4 from the integrand `4*x**2`, resulting in an incorrect coefficient for the resulting integral term. This is a mathematical error in the application of the rule.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.