Integral of \( \displaystyle \frac{1}{9 x + \left(3 x + 1\right)^{2} + 3} \)
Problem 4.1319 · medium Video
Find \( \displaystyle \int \frac{1}{9 x + \left(3 x + 1\right)^{2} + 3} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{9 x + \left(3 x + 1\right)^{2} + 3}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{1}{9 x^{2} + 15 x + 4}\, dx \]algebra simplify algebra complete-squareExpand the squared term. Combine like terms. Factor out the 9. Complete the square for the quadratic part.✓ Proved
- \[ = \int \frac{1}{9 \left(x + \frac{5}{6}\right)^{2} - \frac{9}{4}}\, dx \]algebra simplify algebraDistribute the 9. Simplify the constant term. Write 9/4 as (3/2)**2, then factor out 9 to get (1/2)**2.✓ Proved
- \[ = \int \frac{1}{\left(x + \frac{4}{3}\right) \left(9 x + 3\right)}\, dx \]algebra algebraUse the difference of squares formula. Simplify the terms inside the parentheses.✓ Proved
- \[ = \int \left(- \frac{1}{3 \left(3 x + 4\right)} + \frac{1}{3 \left(3 x + 1\right)}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \int \left(- \frac{1}{9 x + 12} + \frac{1}{9 x + 3}\right)\, dx \]algebraDistribute the 1/3 into the fractions.✓ Proved
- \[ = \frac{\ln{\left(3 x + 1 \right)}}{9} - \frac{\ln{\left(3 x + 4 \right)}}{9} \]antiderivativeIntegrate each term using the log rule.≈ Checked numerically
Answer \( \frac{\ln{\left(x + \frac{1}{3} \right)} - \ln{\left(x + \frac{4}{3} \right)}}{9} + C \)
Lines: 12 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 9*x + (3*x + 1)**2 + 3 = 0 undefined where 9*x**2 + 15*x + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 15*x + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 15*x + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 15*x + 4 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 15*x + 4 = 0 undefined where 9*(x + 5/6)**2 - 9/4 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x + 5/6)**2 - 9/4 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x + 5/6)**2 - 9/4 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x + 5/6)**2 - 9/4 = 0 undefined where x + 4/3 = 0 undefined where 9*x + 3 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 4/3 = 0 undefined where 9*x + 3 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 4/3 = 0 undefined where 9*x + 3 = 0 undefined where 3*x + 1 = 0 undefined where 3*x + 4 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where 3*x + 4 = 0 undefined where 9*x + 12 = 0 undefined where 9*x + 3 = 0 |
| 13 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x + 1/3)/9 - log(x + 4/3)/9 - log(3*x + 1)/9 + log(3*x + 4)/9; numeric agreement only, at 24 of 24 sampled points undefined where 9*x + 12 = 0 undefined where 9*x + 3 = 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 + 1/3)/9 + log(x + 4/3)/9 + log(3*x + 1)/9 - log(3*x + 4)/9; 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 algebraic manipulations, partial fraction decomposition, and integration rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly applies algebraic manipulations, 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 2026-10-23qwen3.6:27b-mlx: inconclusive 2026-10-22 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 11 claims to apply partial fractions but incorrectly transforms the denominator. The term 1/(9*(x+1/3)*(x+4/3)) simplifies to 1/((3x+1)(3gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies algebraic manipulations, 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 2026-10-22
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 2026-10-23 with SymPy 1.14.0.