Integral of \( \displaystyle \ln{\left(2 x + 1 \right)} \)
Problem 4.146 · easy Video
Find \( \displaystyle \int \ln{\left(2 x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \ln{\left(2 x + 1 \right)}\, dx \]integral rewriteStart with the integral of the given function. Rewrite the integrand to prepare for integration by parts.✓ Proved
- \[ = x \ln{\left(2 x + 1 \right)} - \int \frac{2 x}{2 x + 1}\, dx \]parts algebra algebraApply integration by parts with u = log(2*x + 1) and dv = dx. Simplify the integrand in the remaining integral. Rewrite the numerator to facilitate division.✓ Proved
- \[ = x \ln{\left(2 x + 1 \right)} - \int \left(1 - \frac{1}{2 x + 1}\right)\, dx \]algebraSplit the fraction into two parts.✓ Proved
- \[ = x \ln{\left(2 x + 1 \right)} - \int 1\, dx + \int \frac{1}{2 x + 1}\, dx \]linearityDistribute the integral sign.✓ Proved
- \[ = x \ln{\left(2 x + 1 \right)} - x + \int \frac{1}{2 x + 1}\, dx \]algebraSimplify the expression by distributing the negative sign.✓ Proved
- \[ = x \ln{\left(2 x + 1 \right)} - x + 0.5 \ln{\left(2 x + 1 \right)} \]antiderivativeEvaluate the remaining integrals.✓ Proved
Answer \( x \ln{\left(2 x + 1 \right)} - x + \frac{\ln{\left(2 x + 1 \right)}}{2} + C \)
✓ 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 undefined where 2*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 1 = 0 |
| 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 integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed 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 2026-09-29 — The solution correctly applies integration by parts and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.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.