Integral of \( \displaystyle \left(x + 1\right)^{2} \sin{\left(x + 1 \right)} \)
Problem 4.993 · medium
Find \( \displaystyle \int \left(x + 1\right)^{2} \sin{\left(x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(x + 1\right)^{2} \sin{\left(x + 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(x^{2} + 2 x + 1\right) \sin{\left(x + 1 \right)}\, dx \]algebraExpand the squared term.✓ Proved
- \[ = \int 2 x \sin{\left(x + 1 \right)}\, dx + \int x^{2} \sin{\left(x + 1 \right)}\, dx + \int \sin{\left(x + 1 \right)}\, dx \]linearityDistribute the sine term across the polynomial.✓ Proved
- \[ = - \cos{\left(x + 1 \right)} + \int 2 x \sin{\left(x + 1 \right)}\, dx + \int x^{2} \sin{\left(x + 1 \right)}\, dx \]antiderivativeIntegrate the constant term.✓ Proved
- \[ = - 2 x \cos{\left(x + 1 \right)} - \cos{\left(x + 1 \right)} + \int x^{2} \sin{\left(x + 1 \right)}\, dx - \int \left(- 2 \cos{\left(x + 1 \right)}\right)\, dx \]partsApply integration by parts to the second term.✓ Proved
- \[ = - 2 x \cos{\left(x + 1 \right)} - \cos{\left(x + 1 \right)} + \int x^{2} \sin{\left(x + 1 \right)}\, dx + 2 \int \cos{\left(x + 1 \right)}\, dx \]algebraSimplify the expression by distributing the negative sign.✓ Proved
- \[ = - 2 x \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} - \cos{\left(x + 1 \right)} + \int x^{2} \sin{\left(x + 1 \right)}\, dx \]antiderivativeIntegrate the cosine term.✓ Proved
- \[ = - \left(2 x + 1\right) \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} + \int x^{2} \sin{\left(x + 1 \right)}\, dx \]simplifyCombine the cosine terms.✓ Proved
- \[ = - x^{2} \cos{\left(x + 1 \right)} - \left(2 x + 1\right) \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} + \int 2 x \cos{\left(x + 1 \right)}\, dx \]partsApply integration by parts to the first integral term.✓ Proved
- \[ = - x^{2} \cos{\left(x + 1 \right)} + 2 x \sin{\left(x + 1 \right)} - \left(2 x + 1\right) \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} - \int 2 \sin{\left(x + 1 \right)}\, dx \]partsApply integration by parts to the remaining integral term.✓ Proved
- \[ = - x^{2} \cos{\left(x + 1 \right)} + 2 x \sin{\left(x + 1 \right)} - \left(2 x + 1\right) \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} + 2 \cos{\left(x + 1 \right)} \]antiderivativeIntegrate the sine term.✓ Proved
- \[ = - x^{2} \cos{\left(x + 1 \right)} + 2 x \sin{\left(x + 1 \right)} - 2 x \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} \]algebraExpand the parentheses.✓ Proved
- \[ = \left(2 x + 2\right) \sin{\left(x + 1 \right)} + \left(- x^{2} - 2 x + 1\right) \cos{\left(x + 1 \right)} \]simplifyGroup terms by sine and cosine.✓ Proved
Answer \( - x^{2} \cos{\left(x + 1 \right)} + 2 x \sin{\left(x + 1 \right)} - 2 x \cos{\left(x + 1 \right)} + 2 \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} + C \)
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.
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 |
| 13 | ✓ 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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 9 applies integration by parts to the term `Integral(x**2 * sin(x + 1), x)`. The standard integration by parts formula is `uv - Integral(
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 9 applies integration by parts to the term `Integral(x**2 * sin(x + 1), x)`. The standard integration by parts formula is `uv - Integral(qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies integration by parts and algebraic simplification in single-rule steps. The final result matches the stated answer.gpt-oss:20b: pass 2026-10-13
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.