Integral of \( \displaystyle \ln{\left(2 x + 1 \right)} \)
Problem 4.146 · easy
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.
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
Every verdict on record (4)
qwen3.6:27b-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 2026-09-29 with SymPy 1.14.0.