Integral of \( \displaystyle \frac{3 x - 1}{\sqrt{\left(3 x - 1\right)^{2} + 4}} \)
Problem 4.1480 · hard Video
Find \( \displaystyle \int \frac{3 x - 1}{\sqrt{\left(3 x - 1\right)^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x - 1}{\sqrt{\left(3 x - 1\right)^{2} + 4}}\, dx \]integral rewriteStart with the integral of the given function. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = \int \frac{9 x - 3}{3 \sqrt{\left(3 x - 1\right)^{2} + 4}}\, dx \]rewritePrepare for substitution by identifying the derivative of the inner function.✓ Proved
- \[ = \frac{\sqrt{\left(3 x - 1\right)^{2} + 4}}{3} \]antiderivativeThe integral of (3*u/sqrt(u^2+a^2)) is sqrt(u^2+a^2). Here u = 3x-1, so du = 3dx.✓ Proved
Answer \( \frac{\sqrt{\left(3 x - 1\right)^{2} + 4}}{3} + C \)
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 (3*x - 1)**2 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x - 1)**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x - 1)**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: passqwen3.6:27b-mlx: fail (error) — Step 3 applies multiple rules simultaneously (algebraic manipulation to factor out 1/3 and rewrite the numerator as 3*(3x-1)) without a corresponding label for the algebraic changes, violating the one-rule-per-step constraint. Step 4 incorrectly labels the final integration as 'antiderivative' when it relies on a substitution (u = 3x-1) that was not explicitly performed in a dedicated step, and the note describes a substitution rule rather than a direct antiderivative lookup.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-10 — Step 3 applies multiple rules simultaneously (algebraic manipulation to factor out 1/3 and rewrite the numerator as 3*(3x-1)) without a corresponding label for the algebraic changes, violating the one-rule-per-step constraint. Step 4 incorrectly labels the final integration as 'antiderivative' when it relies on a substitution (u = 3x-1) that was not explicitly performed in a dedicated step, and the note describes a substitution rule rather than a direct antiderivative lookup.gpt-oss:20b: pass 2026-11-10qwen3.6:27b-mlx: fail (error) 2026-11-09 — Step 3 applies two rules at once: it rewrites the numerator to introduce a factor of 3 (algebraic manipulation) and factors out 1/3 (linearity) simultaneously. Step 4 is also defective because it jumps from the integral expression to the final answer without explicitly performing the substitution or stating the linearity rule to pull out the constant.gpt-oss:20b: pass 2026-11-09qwen3.6:27b-mlx: fail (error) 2026-11-09 — Step 3 applies multiple rules simultaneously (algebraic manipulation to factor out 1/3 and rewrite the numerator) without a corresponding label like 'algebra' or 'simplify', and the label 'rewrite' is insufficient for the complex transformation. Furthermore, Step 4 jumps directly to the final answer from an incomplete integral expression in Step 3, skipping the actual integration step (substitution) and mislabeling the result as 'antiderivative' rather than showing the integration process.gpt-oss:20b: pass 2026-11-09
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 2026-11-10 with SymPy 1.14.0.