Integral of \( \displaystyle 8 x^{3} + 4 x \)
Problem 4.235 · medium
Find \( \displaystyle \int 8 x^{3} + 4 x \, dx \). (Omit the constant of integration.)
- \[ \int \left(8 x^{3} + 4 x\right)\, dx \]integralStart with the integral of the given polynomial.✓ Proved
- \[ = 4 \int x\, dx + 8 \int x^{3}\, dx \]linearitySplit the integral into two parts using linearity.✓ Proved
- \[ = 2 x^{4} + 2 x^{2} \]antiderivative simplifyApply the power rule to each term. Simplify the coefficients.✓ Proved
Answer \( 2 x^{4} + 2 x^{2} + C \)
✓ 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 |
| 4 | ✓ 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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.