Integral of \( \displaystyle 6 x \sqrt{9 x^{2} + 1} \)
Problem 4.741 · medium
Find \( \displaystyle \int 6 x \sqrt{9 x^{2} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int 6 x \sqrt{9 x^{2} + 1}\, dx \]integral algebra algebraStart with the integral of the given function. Rewrite the square root as a fractional exponent. Rewrite the integrand to reveal the substitution pattern.✓ Proved
- \[ = \frac{2 \left(9 x^{2} + 1\right)^{\frac{3}{2}}}{9} \]antiderivativeIntegrate the expression using the power rule for the substituted form.✓ Proved
Answer \( \frac{2 \left(9 x^{2} + 1\right)^{\frac{3}{2}}}{9} + C \)
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 |
| 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) — Step 4 jumps from the integral expression to the final answer without performing the substitution or applying linearity, violating the 'one change per step' rule. It also mislabels the step as 'antiderivative' when it effectively combines substitution and integration.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 jumps from the integral expression to the final answer without performing the substitution or applying linearity, violating the 'one change per step' rule. It also mislabels the step as 'antiderivative' when it effectively combines substitution and integration.gpt-oss:20b: fail (error) 2026-10-09 — Step 4 applies both a substitution (u=9x^2+1) and the power rule in one line, violating the rule that each step must change only one thing. The label "antiderivative" is insufficient; a "substitution" label is required.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 jumps from the integral expression to the final answer without showing the substitution step or applying linearity to pull out constants. It applies multiple rules (substitution, power rule, constant multiple) at once, violating the one-rule-per-step constraint.
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 2026-10-09 with SymPy 1.14.0.