Integral of \( \displaystyle \frac{1}{9 x^{2} - 4} \)
Problem 4.231 · medium
Find \( \displaystyle \int \frac{1}{9 x^{2} - 4} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{9 x^{2} - 4}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int \frac{1}{\left(3 x - 2\right) \left(3 x + 2\right)}\, dx \]algebraFactor the denominator using the difference of squares.✓ Proved
- \[ = \int \left(- \frac{1}{12 x + 8} + \frac{1}{12 x - 8}\right)\, dx \]partial-fractionsApply partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{12 x - 8}\, dx - \int \frac{1}{12 x + 8}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{\ln{\left(3 x - 2 \right)}}{12} - \frac{\ln{\left(3 x + 2 \right)}}{12} \]antiderivativeEvaluate the integrals using the substitution u = 3x +/- 2.✓ Proved
Answer \( \frac{\ln{\left(x - \frac{2}{3} \right)} - \ln{\left(x + \frac{2}{3} \right)}}{12} + C \)
Lines: 4 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.
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 9*x**2 - 4 = 0 undefined where 3*x + 2 = 0 undefined where 3*x - 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 2 = 0 undefined where 3*x - 2 = 0 undefined where 12*x + 8 = 0 undefined where 12*x - 8 = 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x - 2/3)/12 - log(x + 2/3)/12 - log(12*x - 8)/12 + log(12*x + 8)/12; numeric agreement only, at 24 of 24 sampled points undefined where 12*x + 8 = 0 undefined where 12*x - 8 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 12*x + 8 = 0 undefined where 12*x - 8 = 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 - 2/3)/12 + log(x + 2/3)/12 + log(3*x - 2)/12 - log(3*x + 2)/12; 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: fail (error) — Step 5 applies the antiderivative rule to both terms simultaneously, violating the 'one change per step' constraint. Additionally, the result lacks absolute value signs inside the logarithms, which are required for the general antiderivative of 1/u.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 5 applies the antiderivative rule to both terms simultaneously, violating the 'one change per step' constraint. Additionally, the result lacks absolute value signs inside the logarithms, which are required for the general antiderivative of 1/u.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies partial fraction decomposition and linearity. Step 5 correctly identifies the antiderivatives, and the resulting expression is algebraically equivalent to the stated answer.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.