Home›Calculus 1›Basic antiderivatives›Problem 4.1336

Integral of \( \displaystyle x - 1 + \frac{1}{x - 1} \)

Problem 4.1336 · medium Video

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

\[ \int \left(x - 1 + \frac{1}{x - 1}\right)\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = - \int 1\, dx + \int x\, dx + \int \frac{1}{x - 1}\, dx \]
linearitySplit the integral into three parts using linearity.✓ Proved
\[ = - \int 1\, dx + \int x\, dx + \int \frac{1}{x \left(1 - \frac{1}{x}\right)}\, dx \]
algebraThis step is unnecessary; we will go directly to the antiderivative.✓ Proved
\[ = \frac{x^{2}}{2} - x + \ln{\left(x - 1 \right)} \]
antiderivativeCompute the antiderivative for each term.✓ Proved

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

Top: the graph of F = x²/2 - x + log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = x - 1 + 1/(x - 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 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x - 1 = 0 undefined where x = 0 undefined where 1 - 1/x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where 1 - 1/x = 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: fail (style) — Step 3 applies an unnecessary and confusing algebraic rewrite to only one term, violating the principle of changing only what is needed, and the note admits it is unnecessary. Step 4 then jumps from the rewritten form in Step 3 to the final answer, skipping the actual integration of the intermediate expression, which makes the logical flow disjointed.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-25 — Step 3 applies an unnecessary and confusing algebraic rewrite to only one term, violating the principle of changing only what is needed, and the note admits it is unnecessary. Step 4 then jumps from the rewritten form in Step 3 to the final answer, skipping the actual integration of the intermediate expression, which makes the logical flow disjointed.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 3 applies an unnecessary and complex algebraic rewrite to only one term, violating the 'one thing' principle by introducing a detour that is not used in the subsequent step. Step 4 claims to compute the antiderivative for each term, but the third term in Step 3 is `1/x * (1/(1 - 1/x))`, not `1/(x-1)`, so the label 'antiderivative' does not accurately describe the transition from Step 3 to Step 4 without implying an implicit simplification back to `1/(x-1)` which was not performed.
gpt-oss:20b: pass 2026-10-24
qwen3.6:27b-mlx: pass 2026-10-24
gpt-oss:20b: pass 2026-10-24

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 2026-10-25 with SymPy 1.14.0.