Integral of \( \displaystyle \frac{3 x}{\sqrt{9 x^{2} + 4}} \)
Problem 4.1443 · medium Video
Find \( \displaystyle \int \frac{3 x}{\sqrt{9 x^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x}{\sqrt{9 x^{2} + 4}}\, dx \]integral algebraStart with the integral of the given function. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = \frac{\int \frac{18 x}{\sqrt{9 x^{2} + 4}}\, dx}{6} \]algebraRewrite the numerator to facilitate substitution.✓ Proved
- \[ = \frac{\sqrt{9 x^{2} + 4}}{3} \]antiderivativeApply the power rule for integration.✓ Proved
Answer \( \frac{\sqrt{9 x^{2} + 4}}{3} + C \)
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. 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 undefined where 9*x**2 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 4 = 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 (style) — Step 4 applies two rules (substitution and power rule) but is labeled only as "antiderivative". The correct label should be "substitution" (or "rewrite" followed by "antiderivative").qwen3.6:27b-mlx: pass — The solution correctly applies algebraic manipulation to set up a substitution and then integrates using the power rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly applies algebraic manipulation to set up a substitution and then integrates using the power rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (style) 2026-11-07 — Step 4 applies two rules (substitution and power rule) but is labeled only as "antiderivative". The correct label should be "substitution" (or "rewrite" followed by "antiderivative").qwen3.6:27b-mlx: fail (error) 2026-11-06 — Step 4 applies the power rule for integration directly to the composite function without accounting for the chain rule factor (the derivative of the inner function), resulting in an incorrect coefficient. The step claims to use 'antiderivative' but performs an invalid integration of a composite function.gpt-oss:20b: fail (error) 2026-11-06 — Step 4 applies both a substitution (u=9x^2+4) and an antiderivative, but only the "antiderivative" rule is labeled. Each step must change only one thing and label the rule applied.qwen3.6:27b-mlx: fail (error) 2026-11-06 — Step 4 incorrectly applies the power rule for integration to the composite function (9*x**2 + 4)**(-1/2) without accounting for the chain rule factor or using substitution. The antiderivative of u^(-1/2) is 2*u^(1/2), but since u = 9*x**2 + 4, one must divide by du/dx (or use substitution), which the step ignores, leading to an incorrect intermediate expression before the final answer is somehow reached.gpt-oss:20b: fail (error) 2026-11-06 — Step 4 applies two rules at once (substitution and power rule) and labels the step incorrectly as "antiderivative".
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 2026-11-07 with SymPy 1.14.0.