Home›Calculus 2›Integration by parts›Problem 4.138

Integral of \( \displaystyle \ln{\left(x - 1 \right)} \)

Problem 4.138 · easy Video

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

\[ \int \ln{\left(x - 1 \right)}\, dx \]
integral rewriteStart with the integral of the given function. Rewrite the integrand to prepare for integration by parts.✓ Proved
\[ = x \ln{\left(x - 1 \right)} - \int \frac{x}{x - 1}\, dx \]
parts algebraApply integration by parts with u = log(x - 1) and dv = dx. Rewrite the numerator in the integrand to facilitate division.✓ Proved
\[ = x \ln{\left(x - 1 \right)} - \int \left(1 + \frac{1}{x - 1}\right)\, dx \]
algebraSplit the fraction into two separate terms.✓ Proved
\[ = x \ln{\left(x - 1 \right)} - \int 1\, dx - \int \frac{1}{x - 1}\, dx \]
linearityDistribute the integral sign across the terms.✓ Proved
\[ = x \ln{\left(x - 1 \right)} - x - \ln{\left(x - 1 \right)} \]
antiderivative simplifyEvaluate the integrals of 1 and 1/(x-1). Remove parentheses and simplify the expression.✓ Proved

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

Top: the graph of F = x·log(x - 1) - x - log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = log(x - 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 log is undefined for non-positive arguments
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 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 2026-10-03
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies integration by parts and standard algebraic manipulations. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29

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 2026-10-03 with SymPy 1.14.0.