Integral of \( \displaystyle 3 x \sqrt{1 - 9 x^{2}} \)
Problem 4.1427 · medium
Find \( \displaystyle \int 3 x \sqrt{1 - 9 x^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int 3 x \sqrt{1 - 9 x^{2}}\, dx \]integral simplify algebra substitutionStart with the integral of the given function. Rewrite the square root using exponent notation. Express 9*x**2 as (3*x)**2. Use substitution u = 1 - 9*x**2, where du = -18*x*dx.✓ Proved
- \[ = - \frac{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}}{9} \]antiderivativeEvaluate the integral.✓ Proved
Answer \( - \frac{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}}{9} + C \)
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. 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 |
| 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: fail (error) — Step 4 applies two rules at once: it rewrites the integrand using the substitution and simultaneously introduces the differential du, but the resulting expression is incorrect. The factor –18*x should not appear inside the integral after the substitution; the correct step would be –Integral(1/6*(1-9*x**2)**(1/2) du).qwen3.6:27b-mlx: pass — The solution correctly applies substitution and integration rules. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly applies substitution and integration rules. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-11-04 — Step 4 applies two rules at once: it rewrites the integrand using the substitution and simultaneously introduces the differential du, but the resulting expression is incorrect. The factor –18*x should not appear inside the integral after the substitution; the correct step would be –Integral(1/6*(1-9*x**2)**(1/2) du).qwen3.6:27b-mlx: fail (error) 2026-11-03 — Step 4 applies the substitution rule but fails to change the variable of integration from x to u, leaving the integral in terms of x while introducing du components. Step 5 then incorrectly evaluates the integral without performing the variable change or back-substitution properly, resulting in a mathematically invalid transition.gpt-oss:20b: pass 2026-11-03qwen3.6:27b-mlx: fail (error) 2026-11-03 — Step 4 is not a valid single-step transformation under the 'substitution' rule; it attempts to rewrite the integrand to match a differential form without performing the actual variable change or integration. Furthermore, Step 5 incorrectly applies the antiderivative rule to the original variable expression without showing the substitution back or the integration of the u-form, effectively skipping the integration logic.gpt-oss:20b: fail (error) 2026-11-03 — Step 4 applies two rules at once: it rewrites the integrand using the substitution u=1-9x^2 and simultaneously introduces an extra factor of (-18*x), which is incorrect. The correct substitution yields 3x dx = -du/6, so the integrand becomes -1/6 u^{1/2} du. The step’s label and algebra are therefore incorrect.
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 2026-11-04 with SymPy 1.14.0.