Integral of \( \displaystyle \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}} \)
Problem 4.998 · medium
Find \( \displaystyle \int \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x + 1\right) e^{\left(2 x + 1\right)^{2}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(2.0 x + 1.0\right) e^{\left(2 x + 1\right)^{2}}\, dx \]algebraRewrite the integrand to expose the derivative of the exponent.✓ Proved
- \[ = 0.25 e^{\left(2 x + 1\right)^{2}} \]antiderivativeIntegrate the expression using the substitution rule.✓ Proved
Answer \( \frac{e^{\left(2 x + 1\right)^{2}}}{4} + C \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the derivative of the exponent and uses algebraic manipulation to prepare for substitution, followed by a valid antiderivative step. Each step applies only one rule.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the derivative of the exponent and uses algebraic manipulation to prepare for substitution, followed by a valid antiderivative step. Each step applies only one rule.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the integrand as a composite function times the derivative of its inner function (up to a constant factor). Step 2 uses algebra to adjust coefficients, and Step 3 applies the antiderivative rule for exponential functions. The steps are valid and the final answer is correct.
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 2026-10-13 with SymPy 1.14.0.