Home›Calculus 1›u-substitution›Problem 4.1482

Integral of \( \displaystyle \frac{\ln{\left(2 x + 1 \right)}}{2 x + 1} \)

Problem 4.1482 · medium Video

Find \( \displaystyle \int \frac{\ln{\left(2 x + 1 \right)}}{2 x + 1} \, dx \). (Omit the constant of integration.)

\[ \int \frac{\ln{\left(2 x + 1 \right)}}{2 x + 1}\, dx \]
integral algebra algebra algebra simplify simplifyStart with the integral of the given function. Rewrite the fraction using a negative exponent. Multiply by 1 in the form of 2/2. Rearrange the terms. Simplify the expression. Simplify the constant factor.✓ Proved
\[ = 0.25 \ln{\left(2 x + 1 \right)}^{2} \]
antiderivativeApply the power rule for integration via substitution u = log(2x+1).✓ Proved

Answer \( \frac{\ln{\left(2 x + 1 \right)}^{2}}{4} + C \)

Top: the graph of F = log(2x + 1)²/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = log(2x + 1)/(2x + 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 log is undefined for non-positive arguments undefined where 2*x + 1 = 0
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
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 (error) — Steps 3 through 6 perform redundant algebraic manipulations (multiplying by 1 and rearranging) that do not constitute a single valid rule application in the context of solving the integral, effectively stalling the solution. More critically, Step 7 jumps directly to the final answer without showing the substitution step required by the 'substitution' rule, violating the one-step constraint.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-10 — Steps 3 through 6 perform redundant algebraic manipulations (multiplying by 1 and rearranging) that do not constitute a single valid rule application in the context of solving the integral, effectively stalling the solution. More critically, Step 7 jumps directly to the final answer without showing the substitution step required by the 'substitution' rule, violating the one-step constraint.
gpt-oss:20b: pass 2026-11-10
qwen3.6:27b-mlx: fail (style) 2026-11-09 — Steps 2 through 6 perform redundant algebraic manipulations (multiplying by 1 and rearranging) that do not advance the solution, violating the principle that each step should change one thing meaningfully. Step 7 jumps from the integral to the final answer without showing the substitution step, applying multiple rules (substitution and integration) at once.
gpt-oss:20b: fail (style) 2026-11-09 — Step 4 rearranges the integrand in a way that effectively applies two algebraic manipulations at once (pulling a factor of 2 out and dividing by 2), violating the rule that each step must change only one thing. The rest of the solution is correct.
qwen3.6:27b-mlx: fail (style) 2026-11-09 — Steps 3 through 6 are redundant and do not change the mathematical expression, violating the requirement that each step changes one thing. Step 7 applies substitution and integration in a single step, which should be split into a substitution step and an antiderivative step.
gpt-oss:20b: pass 2026-11-09

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 2026-11-10 with SymPy 1.14.0.