Integral of \( \displaystyle \left(x + 1\right) e^{x + 1} \)
Problem 4.148 · medium Video
Find \( \displaystyle \int \left(x + 1\right) e^{x + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(x + 1\right) e^{x + 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(x e^{x + 1} + e^{x + 1}\right)\, dx \]algebraDistribute the term (x + 1).✓ Proved
- \[ = \int x e^{x + 1}\, dx + \int e^{x + 1}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = e^{x + 1} + \int x e^{x + 1}\, dx \]antiderivative partsIntegrate the second term. Apply integration by parts to the first integral term.✓ Proved
- \[ = x e^{x + 1} + e^{x + 1} - \int e^{x + 1}\, dx \]parts simplifyEvaluate the parts formula for the first integral. Cancel the redundant integral terms.✓ Proved
- \[ = x e^{x + 1} \]antiderivative simplifyIntegrate the remaining integral term. Combine like terms to find the final antiderivative.✓ Proved
Answer \( x e^{x + 1} + C \)
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 5 applies two operations at once: it adds and subtracts an integral while claiming to perform integration by parts. This violates the rule that each step must change only one thing. The subsequent parts steps (6–8) also misuse the integration‑by‑parts formula, leading to an incorrect manipulation of the integral. The solution therefore contains a mathematical error.qwen3.6:27b-mlx: fail (error) — Step 5 is algebraically invalid; it adds and subtracts an integral without justification, and Step 6 misapplies the integration by parts formula (the derivative of x is 1, not exp(x+1), so the remaining integral should not be multiplied by exp(x+1) in the integrand). The solution fabricates terms to force the answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 is algebraically invalid; it adds and subtracts an integral without justification, and Step 6 misapplies the integration by parts formula (the derivative of x is 1, not exp(x+1), so the remaining integral should not be multiplied by exp(x+1) in the integrand). The solution fabricates terms to force the answer.gpt-oss:20b: fail (error) 2026-10-03 — Step 5 applies two operations at once: it adds and subtracts an integral while claiming to perform integration by parts. This violates the rule that each step must change only one thing. The subsequent parts steps (6–8) also misuse the integration‑by‑parts formula, leading to an incorrect manipulation of the integral. The solution therefore contains a mathematical error.qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is labeled 'parts' but performs an algebraic addition of zero (adding and subtracting the same integral), which is not integration by parts. Step 6 is labeled 'parts' but performs algebraic simplification of the terms added in Step 5, not the application of the integration by parts formula.gpt-oss:20b: fail (error) 2026-09-29 — Step 5 applies two operations at once: it adds and subtracts the same integral (creating a zero) and then attempts integration by parts. This violates the rule that a step must change only one thing and be labeled with a single rule. The correct approach would apply integration by parts directly to ∫ x e^{x+1} dx without inserting an extra zero term.qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is mathematically incorrect; adding and subtracting the same integral is an algebraic manipulation, not the application of integration by parts. Step 6 then incorrectly applies the integration by parts formula to the first term while simultaneously performing unrelated cancellations, violating the one-rule-per-step constraint and producing a logically incoherent derivation.gpt-oss:20b: fail (error) 2026-09-29 — Step 5 adds and subtracts the same integral in one line, which is two separate changes, and it is labeled as "parts" even though no integration‑by‑parts formula is applied. This violates the one‑rule‑change rule and mislabels the operation.
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.