Integral of \( \displaystyle 2 x + 1 + \frac{1}{2 x + 1} \)
Problem 4.879 · medium
Find \( \displaystyle \int 2 x + 1 + \frac{1}{2 x + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \left(2 x + 1 + \frac{1}{2 x + 1}\right)\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int 1\, dx + \int 2 x\, dx + \int \frac{1}{2 x + 1}\, dx \]linearitySplit the integral into three parts.✓ Proved
- \[ = x^{2} + x + \int \frac{1}{2 x + 1}\, dx \]antiderivative algebraIntegrate the polynomial terms. Factor out the 2 from the denominator.✓ Proved
- \[ = x^{2} + x + \frac{\int \frac{1}{x + \frac{1}{2}}\, dx}{2} \]rewriteMove the constant factor outside the integral.✓ Proved
- \[ = x^{2} + x + \frac{\ln{\left(2 x + 1 \right)}}{2} \]antiderivativeIntegrate the remaining term using the substitution result.✓ Proved
Answer \( x^{2} + x + \frac{\ln{\left(2 x + 1 \right)}}{2} + C \)
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.
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 undefined where 2*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 undefined where x + 1/2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/2 = 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: fail (error) — Step 6 integrates 1/(x+1/2) to log(2*x+1) which is incorrect; the correct antiderivative is log(x+1/2), so the final result should be x**2 + x + (1/2)*log(x+1/2).qwen3.6:27b-mlx: pass — The solution correctly applies linearity to split the integral, integrates polynomial terms, and handles the rational term via algebraic manipulation and substitution. All steps adhere to the single-rule constraint and use valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies linearity to split the integral, integrates polynomial terms, and handles the rational term via algebraic manipulation and substitution. All steps adhere to the single-rule constraint and use valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-10-11 — Step 6 integrates 1/(x+1/2) to log(2*x+1) which is incorrect; the correct antiderivative is log(x+1/2), so the final result should be x**2 + x + (1/2)*log(x+1/2).qwen3.6:27b-mlx: pass 2026-10-11gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.