Integral of \( \displaystyle \frac{2 x + 1}{x^{2} + x + 3} \)
Problem 4.217 · medium
Find \( \displaystyle \int \frac{2 x + 1}{x^{2} + x + 3} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x + 1}{x^{2} + x + 3}\, dx \]integral rewriteStart with the given integral. Notice that the numerator is the derivative of the denominator.✓ Proved
- \[ = \ln{\left(x^{2} + x + 3 \right)} \]antiderivativeThe integral of u'/u is log(u).✓ Proved
Answer \( \ln{\left(x^{2} + x + 3 \right)} + C \)
✓ 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 undefined where x**2 + x + 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + x + 3 = 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 identifies the numerator as the derivative of the denominator and applies the standard logarithmic integration rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the numerator as the derivative of the denominator and applies the standard logarithmic integration rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the numerator as the derivative of the denominator and applies the standard logarithmic integration rule. Each step changes only one aspect of the expression and uses valid labels.gpt-oss:20b: pass 2026-10-03
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.