Integral of \( \displaystyle x^{2} e^{x^{3}} \)
Problem 4.230 · medium
Find \( \displaystyle \int x^{2} e^{x^{3}} \, dx \). (Omit the constant of integration.)
- \[ \int x^{2} e^{x^{3}}\, dx \]integral algebra algebra algebra simplify algebra algebra substitutionStart with the integral of the given function. Multiply and divide by the derivative of the exponent, 3x^2. Group the terms for clarity. Simplify the fraction (3*x^2)/(3*x^2) to 1, but keep the structure to show the substitution factor. The expression simplifies back to the original integrand. Rewrite the integrand to explicitly show the derivative of the inner function. Simplify the constant factor. The integrand is now in the form (1/3) * d/dx(exp(x**3)).✓ Proved
- \[ = \frac{e^{x^{3}}}{3} \]antiderivativeApply the antiderivative rule for the exponential function.✓ Proved
Answer \( \frac{e^{x^{3}}}{3} + C \)
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 |
| 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: fail (error) — Steps 2 through 7 are circular and do not constitute a valid derivation; they merely rearrange the integrand back to its original form without performing the substitution. Step 8 claims to apply substitution but skips the actual variable change (u = x^3) and differential adjustment, jumping directly from the original integrand to the result.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Steps 2 through 7 are circular and do not constitute a valid derivation; they merely rearrange the integrand back to its original form without performing the substitution. Step 8 claims to apply substitution but skips the actual variable change (u = x^3) and differential adjustment, jumping directly from the original integrand to the result.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution contains a logical loop where steps 5, 6, and 7 undo the progress of steps 2-4, returning to the original integrand. Furthermore, step 8 claims to apply 'substitution' but performs no algebraic change, and step 9 jumps from the integral of the original function to the answer without showing the actual substitution variable change or integration.gpt-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.