Integral of \( \displaystyle \ln{\left(2 x \right)} \)
Problem 4.154 · easy Video
Find \( \displaystyle \int \ln{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \ln{\left(2 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(\ln{\left(x \right)} + \ln{\left(2 \right)}\right)\, dx \]algebraUse the logarithm product rule to split the term.✓ Proved
- \[ = \int \ln{\left(2 \right)}\, dx + \int \ln{\left(x \right)}\, dx \]linearitySplit the integral into two separate integrals.✓ Proved
- \[ = x \ln{\left(2 \right)} + \int \ln{\left(x \right)}\, dx \]antiderivativeThe integral of a constant is the constant times x.✓ Proved
- \[ = x \ln{\left(x \right)} + x \ln{\left(2 \right)} - \int 1\, dx \]partsUse integration by parts on the integral of log(x) with u=log(x) and dv=dx.✓ Proved
- \[ = x \ln{\left(x \right)} - x + x \ln{\left(2 \right)} \]antiderivativeEvaluate the integral of 1 with respect to x.✓ Proved
Answer \( x \ln{\left(2 x \right)} - x + C \)
✓ 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies logarithm properties, linearity of integration, and integration by parts. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies logarithm properties, linearity of integration, and integration by parts. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29gpt-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.