Integral of \( \displaystyle 4 x^{2} \sin{\left(2 x \right)} \)
Problem 4.165 · medium Video
Find \( \displaystyle \int 4 x^{2} \sin{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int 4 x^{2} \sin{\left(2 x \right)}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = 4 \int x^{2} \sin{\left(2 x \right)}\, dx \]linearityFactor out the constant 4.✓ Proved
- \[ = - 2 x^{2} \cos{\left(2 x \right)} + 4 \int x \cos{\left(2 x \right)}\, dx \]parts algebraApply integration by parts with u = x**2 and dv = sin(2*x)dx. Add parentheses for clarity.✓ Proved
- \[ = - 2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} - 4 \int \frac{\sin{\left(2 x \right)}}{2}\, dx \]partsApply integration by parts again to the remaining integral.✓ Proved
- \[ = - 2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \]antiderivative simplify simplifyEvaluate the integral of sin(2*x). Simplify the signs and fractions. Distribute the 4 to get the final antiderivative.✓ Proved
Answer \( - 2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + C \)
✓ 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. 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 |
| 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 applies integration by parts twice and handles algebraic simplification properly. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies integration by parts twice and handles algebraic simplification properly. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts twice and handles the constant factors and signs accurately. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.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.