Integral of \( \displaystyle \frac{4 x + 3}{2 x + \left(2 x + 1\right)^{2} + 4} \)
Problem 4.994 · hard
Find \( \displaystyle \int \frac{4 x + 3}{2 x + \left(2 x + 1\right)^{2} + 4} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{4 x + 3}{2 x + \left(2 x + 1\right)^{2} + 4}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{4 x + 3}{4 x^{2} + 6 x + 5}\, dx \]algebra simplify rewriteExpand the squared term. Combine like terms in the denominator. Rewrite the numerator to match the derivative of the denominator.✓ Proved
- \[ = \frac{\ln{\left(4 x^{2} + 6 x + 5 \right)}}{2} \]antiderivativeIntegrate using the logarithmic rule.✓ Proved
Answer \( \frac{\ln{\left(4 x^{2} + 6 x + 5 \right)}}{2} + C \)
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 + (2*x + 1)**2 + 4 = 0 undefined where 4*x**2 + 6*x + 5 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 6*x + 5 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 6*x + 5 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 6*x + 5 = 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 simplifies the denominator and identifies the numerator as a constant multiple of the denominator's derivative, applying 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)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly simplifies the denominator and identifies the numerator as a constant multiple of the denominator's derivative, applying the standard logarithmic integration rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-10-13 — Step 4 rewrites the integrand as (1/2*(8*x+6))/(4*x**2+6*x+5), which equals (8*x+6)/(2*(4*x**2+6*x+5)). This is not algebraically equivalent to the original integrand (4*x+3)/(4*x**2+6*x+5). Consequently, the subsequent antiderivative in step 5 is based on an incorrect integrand.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly simplifies the denominator and identifies the numerator as proportional to the derivative of the denominator, applying the standard logarithmic integration rule. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
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 2026-10-13 with SymPy 1.14.0.