Home›Calculus 1›u-substitution›Problem 4.1217

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

Problem 4.1217 · medium Video

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

\[ \int \frac{3 x - 1}{\left(3 x - 1\right)^{2} + 1}\, dx \]
integral rewriteStart with the integral of the given function. Rewrite the numerator to include the derivative of the denominator.✓ Proved
\[ = \frac{\int \frac{18 x - 6}{\left(3 x - 1\right)^{2} + 1}\, dx}{6} \]
linearity algebraFactor out the constant 1/6. Rearrange the numerator to clearly show the derivative of the denominator.✓ Proved
\[ = \frac{\ln{\left(\left(3 x - 1\right)^{2} + 1 \right)}}{6} \]
antiderivativeApply the antiderivative rule for the form u'/u.✓ Proved

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

Top: the graph of F = log(9x² - 6x + 2)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (3x - 1)/((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 undefined where (3x - 1)*2 + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x - 1)*2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x - 1)*2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (3x - 1)*2 + 1 = 0 log is undefined for non-positive arguments
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies substitution logic via algebraic rewriting and linearity, with appropriate labels for each single-step transformation.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies substitution via algebraic manipulation and linearity, followed by the standard antiderivative rule for u'/u. Each step changes only one aspect of the expression and uses valid labels.
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.