Home›Calculus 1›u-substitution›Problem 4.1354

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

Problem 4.1354 · medium Video

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

\[ \int \frac{2 x - 1}{x + \left(x - 1\right)^{2} + 2}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{2 x - 1}{x^{2} - x + 3}\, dx \]
algebra simplify rewrite algebraExpand the squared term. Combine like terms in the denominator. Recognize that the numerator is the derivative of the denominator. Compute the derivative of the denominator.✓ Proved
\[ = \ln{\left(x^{2} - x + 3 \right)} \]
antiderivativeIntegrate the expression using the logarithmic rule.✓ Proved

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

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

✓ 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 x + (x - 1)2 + 2 = 0 undefined where x2 - x + 3 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - x + 3 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - x + 3 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - x + 3 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - x + 3 = 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: fail (style) — Step 5 applies the derivative rule to replace diff(x**2 - x + 3, x) with 2*x - 1, but it is labeled as "algebra". The correct label for this step is "derivative".
qwen3.6:27b-mlx: pass — The solution correctly simplifies the denominator and identifies the numerator as the derivative of the denominator, leading to the correct logarithmic antiderivative. All steps follow the single-change rule and use valid labels.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly simplifies the denominator and identifies the numerator as the derivative of the denominator, leading to the correct logarithmic antiderivative. All steps follow the single-change rule and use valid labels.
gpt-oss:20b: fail (style) 2026-10-28 — Step 5 applies the derivative rule to replace diff(x**2 - x + 3, x) with 2*x - 1, but it is labeled as "algebra". The correct label for this step is "derivative".
qwen3.6:27b-mlx: pass 2026-10-27
gpt-oss:20b: pass 2026-10-27
qwen3.6:27b-mlx: pass 2026-10-27
gpt-oss:20b: fail (style) 2026-10-27 — Step 5 recomputes the derivative of the denominator but labels it as "algebra". The correct rule for computing a derivative is "derivative", not "algebra". This mislabeling violates the granularity requirement.

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 2026-10-28 with SymPy 1.14.0.