Integral of \( \displaystyle \frac{1}{x^{2} + 3 x} \)
Problem 4.191 · medium
Find \( \displaystyle \int \frac{1}{x^{2} + 3 x} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{x^{2} + 3 x}\, dx \]integralStart with the given integral.✓ Proved
- \[ = \int \frac{1}{x \left(x + 3\right)}\, dx \]algebraFactor the denominator.✓ Proved
- \[ = \int \left(- \frac{1}{3 x + 9} + \frac{1}{3 x}\right)\, dx \]partial-fractionsPerform partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{3 x}\, dx - \int \frac{1}{3 x + 9}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{\int \frac{1}{x}\, dx}{3} - \frac{\int \frac{1}{x + 3}\, dx}{3} \]linearityFactor out the constant 1/3.✓ Proved
- \[ = \frac{\ln{\left(x \right)}}{3} - \frac{\ln{\left(x + 3 \right)}}{3} \]antiderivativeIntegrate each term using the natural logarithm rule.✓ Proved
Answer \( \frac{\ln{\left(x \right)} - \ln{\left(x + 3 \right)}}{3} + C \)
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 x**2 + 3*x = 0 undefined where x + 3 = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0 undefined where 3*x + 9 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | lines differ by the constant log(3)/3 undefined where 3*x + 9 = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -log(3)/3 undefined where 3*x + 9 = 0 undefined where x = 0 undefined where x + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 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: fail (misleading) — The solution omits the absolute value in the logarithms, which would mislead a student into believing ∫1/x dx = log(x) instead of log|x|.qwen3.6:27b-mlx: pass — The solution correctly applies partial fraction decomposition, linearity of integration, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies partial fraction decomposition, linearity of integration, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (misleading) 2026-09-29 — The solution omits the absolute value in the logarithms, which would mislead a student into believing ∫1/x dx = log(x) instead of log|x|.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies partial fraction decomposition, linearity, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.