Integral of \( \displaystyle \frac{1}{4 x^{2} + 6 x} \)
Problem 4.1216 · medium Video
Find \( \displaystyle \int \frac{1}{4 x^{2} + 6 x} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{4 x^{2} + 6 x}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int \frac{1}{2 x \left(2 x + 3\right)}\, dx \]algebra algebra algebraFactor the denominator. Prepare for partial fraction decomposition by introducing a constant multiplier. Rewrite the numerator 3 as (2*x + 3) - 2*x.✓ Proved
- \[ = \int \left(- \frac{1}{3 \left(2 x + 3\right)} + \frac{1}{6 x}\right)\, dx \]algebra simplifyDistribute the terms in the numerator. Simplify each fraction by canceling common terms.✓ Proved
- \[ = \int \frac{1}{6 x}\, dx - \int \frac{1}{6 x + 9}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\ln{\left(x \right)}}{6} - \int \frac{1}{6 x + 9}\, dx \]antiderivativeIntegrate the first term.✓ Proved
- \[ = \frac{\ln{\left(x \right)}}{6} - \frac{\int \frac{1}{2 x + 3}\, dx}{3} \]linearityFactor out the constant from the second integral.✓ Proved
- \[ = \frac{\ln{\left(x \right)}}{6} - \frac{\ln{\left(2 x + 3 \right)}}{6} \]substitution algebra algebraIntegrate the remaining term using u-substitution. Multiply the constants. Factor out 1/6.✓ Proved
Answer \( \frac{\ln{\left(x \right)} - \ln{\left(x + \frac{3}{2} \right)}}{6} + C \)
Lines: 11 proved, 2 checked numerically. 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 undefined where 4*x**2 + 6*x = 0 undefined where x = 0 undefined where 2*x + 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 2*x + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 2*x + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 2*x + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 2*x + 3 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -log(x + 3/2)/6 + log(6*x + 9)/6; numeric agreement only, at 24 of 24 sampled points undefined where x = 0 undefined where 2*x + 3 = 0 undefined where 6*x + 9 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6*x + 9 = 0 undefined where x = 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -log(3)/6 log is undefined for non-positive arguments undefined where 6*x + 9 = 0 undefined where 2*x + 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left log(x + 3/2)/6 - log(2*x + 3)/6; numeric agreement only, at 24 of 24 sampled points 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 applies partial fraction decomposition and integration rules. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies partial fraction decomposition and integration rules. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies partial fraction decomposition and integration rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The final answer contains log(2*x + 3) instead of log(x + 3/2). While these differ by a constant (log(2)), the problem statement explicitly provides the target answer as (log(x) - log(x + 3/2))/6. The solution fails to simplify log(2*x + 3) to log(x + 3/2) + log(2) and absorb the constant, resulting in a mismatch with the stated answer.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.