Home›Calculus 1›u-substitution›Problem 4.1408

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

Problem 4.1408 · hard Video

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

\[ \int \left(6 x - 2\right) \sqrt{\left(3 x - 1\right)^{2} + 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(6 x - 2\right) \sqrt{9 x^{2} - 6 x + 2}\, dx \]
algebra algebraExpand the term inside the square root. Rewrite the integrand to reveal the derivative of the inner function.✓ Proved
\[ = \frac{2 \left(9 x^{2} - 6 x + 2\right)^{\frac{3}{2}}}{9} \]
antiderivativeIntegrate using the power rule and chain rule.✓ Proved

Answer \( \frac{2 \left(9 x^{2} - 6 x + 2\right)^{\frac{3}{2}}}{9} + C \)

Top: the graph of F = 2(9x² - 6x + 2)^(3/2)/9. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (6x - 2)·sqrt((3x - 1)² + 1).

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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
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: pass
qwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'antiderivative' but performs the integration in a single step without showing the substitution or power rule application, violating the 'one thing per step' constraint. It should be split into a substitution step and an integration step.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-11-02 — Step 4 is labeled 'antiderivative' but performs the integration in a single step without showing the substitution or power rule application, violating the 'one thing per step' constraint. It should be split into a substitution step and an integration step.
gpt-oss:20b: pass 2026-11-02
qwen3.6:27b-mlx: fail (style) 2026-11-01 — Step 3 applies two algebraic manipulations simultaneously: factoring out 1/3 and rewriting 2*(3x-1) as (18x-6)/3. Step 4 uses the label 'antiderivative' but the note mentions 'chain rule', which is not in the allowed vocabulary and implies a differentiation step rather than integration by substitution.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies substitution via algebraic manipulation and the power rule for integration. Each step changes only one aspect of the expression, and the labels are appropriate.
gpt-oss:20b: pass 2026-11-01

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 2026-11-02 with SymPy 1.14.0.