Integral of \( \displaystyle \left(2 x + 1\right)^{2} e^{2 x + 1} \)
Problem 4.1282 · 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
- \[ = \int \left(4 x^{2} e^{2 x + 1} + 4 x e^{2 x + 1} + e^{2 x + 1}\right)\, dx \]algebraExpand the squared term.✓ Proved
- \[ = \int 4 x e^{2 x + 1}\, dx + \int 4 x^{2} e^{2 x + 1}\, dx + \int e^{2 x + 1}\, dx \]linearitySplit into three separate integrals.✓ Proved
- \[ = \left(2 x - 1\right) e^{2 x + 1} + \frac{e^{2 x + 1}}{2} + \int 4 x^{2} e^{2 x + 1}\, dx \]antiderivativeIntegrate the second and third terms.✓ Proved
- \[ = \left(2 x - 1\right) e^{2 x + 1} + \left(2 x^{2} - 2 x + 1\right) e^{2 x + 1} + \frac{e^{2 x + 1}}{2} \]partsIntegrate the first term using integration by parts.✓ Proved
- \[ = \left(2 x^{2} + \frac{1}{2}\right) e^{2 x + 1} \]simplifyCombine all terms and simplify.✓ Proved
Answer \( \frac{\left(4 x^{2} + 1\right) e^{2 x + 1}}{2} + C \)
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 |
| 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 4 applies the antiderivative rule to two separate terms simultaneously, violating the constraint that each step must change only one thing. Additionally, Step 5 incorrectly labels the integration of the remaining term as 'parts' when the previous step already performed the integration, making the label in Step 5 refer to a step that is actually just algebraic substitution or simplification, not a new integration rule application.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 4 applies the antiderivative rule to two separate terms simultaneously, violating the constraint that each step must change only one thing. Additionally, Step 5 incorrectly labels the integration of the remaining term as 'parts' when the previous step already performed the integration, making the label in Step 5 refer to a step that is actually just algebraic substitution or simplification, not a new integration rule application.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies algebraic expansion, linearity of integration, and integration by parts. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly expands the integrand, applies linearity, and integrates each term. The steps adhere to the one-change-per-step rule, and the labels accurately describe the operations performed.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.