Home›Calculus 1›u-substitution›Problem 4.1128

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

Problem 4.1128 · medium Video

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

\[ \int \frac{x - 1}{\left(x - 1\right)^{2} + 1}\, dx \]
integralStart with the given integral.✓ Proved
\[ = \int \left(\frac{x}{x^{2} - 2 x + 2} - \frac{1}{x^{2} - 2 x + 2}\right)\, dx \]
algebraSplit the fraction into two parts.✓ Proved
\[ = \int \frac{x}{x^{2} - 2 x + 2}\, dx - \int \frac{1}{x^{2} - 2 x + 2}\, dx \]
linearity algebraSeparate the integral into two terms. Rewrite the numerator to involve the derivative of the denominator.✓ Proved
\[ = \int \left(\frac{x - 1}{x^{2} - 2 x + 2} + \frac{1}{x^{2} - 2 x + 2}\right)\, dx - \int \frac{1}{x^{2} - 2 x + 2}\, dx \]
algebraDistribute the 1/2.✓ Proved
\[ = \int \frac{x - 1}{x^{2} - 2 x + 2}\, dx \]
linearity simplifySplit the first integral. The two identical integrals cancel out.✓ Proved
\[ = \frac{\ln{\left(x^{2} - 2 x + 2 \right)}}{2} \]
antiderivativeIntegrate using the substitution u = x^2 - 2x + 2.✓ Proved

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

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

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 (x - 1)2 + 1 = 0 undefined where x2 - 2*x + 2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x + 2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x + 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x + 2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x + 2 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x + 2 = 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 — The solution correctly applies algebraic manipulation and linearity to simplify the integrand, and the final integration step is valid. The steps adhere to the one-change-per-step rule and use appropriate labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies algebraic manipulation and linearity to simplify the integrand, and the final integration step is valid. The steps adhere to the one-change-per-step rule and use appropriate labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies algebraic manipulation to rewrite the numerator as the derivative of the denominator, followed by linearity and simplification. The final integration step is correctly labeled and executed.
gpt-oss:20b: fail (error) 2026-10-15 — Step 5 applies two operations at once: it distributes the 1/2 over the sum and simultaneously rewrites the fraction, violating the one‑rule‑per‑step rule. The correct distribution would give Integral(x/(x**2-2*x+2),x), not the expression shown.
qwen3.6:27b-mlx: pass 2026-10-14
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.