Home›Calculus 1›u-substitution›Problem 4.1220

Integral of \( \displaystyle \frac{6 x - 1}{3 x + \left(3 x - 1\right)^{2} + 2} \)

Problem 4.1220 · hard Video

Find \( \displaystyle \int \frac{6 x - 1}{3 x + \left(3 x - 1\right)^{2} + 2} \, dx \). (Omit the constant of integration.)

\[ \int \frac{6 x - 1}{3 x + \left(3 x - 1\right)^{2} + 2}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{6 x - 1}{9 x^{2} - 3 x + 3}\, dx \]
algebra simplify rewriteExpand the squared term. Combine like terms in the denominator. Rewrite the numerator to match the derivative of the denominator.✓ Proved
\[ = \frac{\ln{\left(9 x^{2} - 3 x + 3 \right)}}{3} \]
antiderivativeIntegrate using the substitution u = 9x^2 - 3x + 3.✓ Proved

Answer \( \frac{\ln{\left(3 x^{2} - x + 1 \right)}}{3} + C \)

Top: the graph of F = log(3x² - x + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (6x - 1)/(3x + (3x - 1)² + 2).

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 3x + (3x - 1)*2 + 2 = 0 undefined where 9x*2 - 3x + 3 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 9x2 - 3x + 3 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 9x2 - 3x + 3 = 0
5	✓ Proved	sympy 1.14.0	lines differ by the constant -log(3)/3 undefined where 9x2 - 3x + 3 = 0 log is undefined for non-positive arguments
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: lines differ by the constant log(3)/3 log is undefined for non-positive arguments
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated the stated antiderivative back to the integrand

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly simplifies the denominator and identifies the numerator as a constant multiple of the denominator's derivative. The final result is algebraically equivalent to the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly simplifies the denominator and identifies the numerator as a constant multiple of the denominator's derivative. The final result is algebraically equivalent to the stated answer.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 4 applies two operations simultaneously: it factors out 1/3 from the numerator AND modifies the remaining numerator from (6x-1) to (18x-3). This violates the 'one rule per step' constraint. Additionally, the note in Step 5 claims a substitution was used, but the step label is 'antiderivative' and the transition from Step 4 to 5 skips the explicit substitution step or the recognition that the numerator is not exactly the derivative of the denominator (derivative of 9x^2-3x+3 is 18x-3, so the numerator 18x-3 matches, but the factor 1/3 was pulled out in step 4, making the integrand 1/3 * (18x-3)/(denom). Wait, let's re-read carefully. Step 4 rewrites (6x-1) as 1/3*(18x-3). This is algebra. Step 5 integrates. The derivative of 9x^2-3x+3 is 18x-3. So the integral of (18x-3)/(9x^2-3x+3) is log(9x^2-3x+3). The 1/3 factor is carried through. So the math is correct. The defect is purely that Step 4 does two things: it introduces the factor 1/3 AND changes the polynomial inside. Actually, rewriting (6x-1) as 1/3*(18x-3) is a single algebraic manipulation. Is it? 6x-1 = 1/3 * 3 * (6x-1) = 1/3 * (18x-3). This is one algebraic step. So Step 4 is fine. Let's look at Step 5. It goes from Integral(1/3 * (18x-3)/(9x^2-3x+3), x) to 1/3 * log(9x^2-3x+3). This applies the antiderivative rule for log(u)/u' and the constant multiple rule. The label is 'antiderivative'. The note says 'using substitution'. This is acceptable. Is there a defect? Let's check the final answer. The stated answer is log(3*x**2 - x + 1)/3. The solution ends with 1/3 * log(9*x**2 - 3*x + 3). Note that 9x^2-3x+3 = 3(3x^2-x+1). So log(9x^2-3x+3) = log(3) + log(3x^2-x+1). The constant log(3)/3 is absorbed into the constant of integration. So the final expression is equivalent. However, the solution stops at 1/3 * log(9*x**2 - 3*x + 3). The 'Stated answer' is log(3*x**2 - x + 1)/3. These differ by a constant. The prompt says 'the constant of integration is omitted throughout'. This usually implies we are looking for a specific antiderivative or that constants don't matter. But does the solution match the stated answer? The solution's final line is 1/3 * log(9*x**2 - 3*x + 3). The stated answer is log(3*x**2 - x + 1)/3. They are not identical expressions. The solver failed to simplify the argument of the logarithm to match the stated answer format, or rather, the stated answer is a simplified version. Is this a defect? The contract says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. It does not say the final line must equal the stated answer exactly, but usually 'Stated answer' is the target. If the final line is not the stated answer, is it a fail? The prompt asks to review the solution. If the solution ends with an expression that is not the stated answer, but is equivalent up to a constant, is that a defect? In many contexts, yes, because the 'answer' is a specific form. However, the instructions say 'find what is wrong... not to be agreeable'. Let's look closer at Step 4. (6x-1) -> 1/3*(18x-3). This is valid. Step 5. Integral(1/3 * (18x-3)/(9x^2-3x+3)) -> 1/3 * log(9x^2-3x+3). This is valid. The result is correct. The only potential issue is the mismatch with the 'Stated answer'. But wait, look at Step 2. (3*x - 1)**2 expands to 9x^2 - 6x + 1. The denominator is 3x + 9x^2 - 6x + 1 + 2 = 9x^2 - 3x + 3. This is correct. Step 3 combines terms. Correct. Step 4 rewrites numerator. Correct. Step 5 integrates. Correct. The final result 1/3 log(9x^2-3x+3) is a valid antiderivative. The stated answer 1/3 log(3x^2-x+1) is also a valid antiderivative. They differ by 1/3 log(3). Since constants are omitted, they are equivalent. Is there a defect in the steps? Step 4 label 'rewrite'. Step 5 label 'antiderivative'. The note in Step 5 says 'using substitution u = ...'. This is a valid explanation. I don't see a defect in the steps themselves. The only 'defect' might be that the final line doesn't match the stated answer's form, but that's often acceptable in indefinite integration. However, looking at the strictness of 'adversarial' review: Did Step 4 apply two rules? It multiplied by 1 and divided by 3? No, it just factored. Did Step 5 apply two rules? It applied the constant multiple rule (pulling 1/3 out) and the antiderivative rule. The label is 'antiderivative'. The constant multiple rule is often considered part of 'linearity' or 'algebra'. If the step pulls out the constant AND integrates, that's two rules. The step goes from `Integral(1/3 * ...)` to `1/3 * log(...)`. This involves recognizing the integral of the function part AND moving the constant outside. If 'antiderivative' covers the integration of the core function, does it cover the constant multiple? Usually, `Integral(c * f) = c * Integral(f)` is linearity. Then `Integral(f) = F` is antiderivative. Step 5 does both. It transforms the integral expression into the evaluated expression. This is a composite step. It applies linearity (constant multiple) and antiderivative. The label is only 'antiderivative'. This is a defect: a step applying two rules (linearity and antiderivative) labeled with only one. Or, if 'antiderivative' is interpreted as 'evaluate the integral', it might be acceptable. But the contract says 'Each step changes ONE thing'. Changing `Integral(c * f)` to `c * F` changes the integral sign, the constant position, and the function to its antiderivative. That's more than one thing. A proper sequence would be: 1. Pull out 1/3 (linearity). 2. Integrate the rest (antiderivative). Step 5 does both. Therefore, it violates the one-rule-per-step constraint.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 4 violates the single-rule constraint by simultaneously factoring out 1/3 and rewriting the remaining numerator (6x-1 to 18x-3) to match the derivative of the denominator. Step 5 is also defective because it applies the antiderivative rule without explicitly performing the substitution step required to justify the result.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.