Integral of \( \displaystyle \frac{x - 1}{x \left(x + 2\right)} \)
Problem 4.1337 · medium Video
Find \( \displaystyle \int \frac{x - 1}{x \left(x + 2\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{x - 1}{x \left(x + 2\right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{x - 1}{x^{2} + 2 x}\, dx \]algebraExpand the denominator.✓ Proved
- \[ = \int \left(\frac{1}{x + 2} - \frac{1}{x \left(x + 2\right)}\right)\, dx \]algebra algebra algebra algebraSplit the numerator into two terms. Rewrite the second term. Isolate the term 1/x. Prepare for partial fraction decomposition.✓ Proved
- \[ = \int \left(\frac{1}{2 x + 4} + \frac{1}{x + 2} - \frac{1}{2 x}\right)\, dx \]algebraDistribute the term 1/x.✓ Proved
- \[ = \int \left(\frac{3}{2 x + 4} - \frac{1}{2 x}\right)\, dx \]algebraCombine like terms.✓ Proved
- \[ = - \int \frac{1}{2 x}\, dx + \int \frac{3}{2 x + 4}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = - \frac{\ln{\left(x \right)}}{2} + \frac{3 \ln{\left(x + 2 \right)}}{2} \]antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{- \ln{\left(x \right)} + 3 \ln{\left(x + 2 \right)}}{2} + C \)
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. 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 x = 0 undefined where x + 2 = 0 undefined where x**2 + 2*x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 2*x = 0 undefined where x = 0 undefined where x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where x + 2 = 0 undefined where 2*x + 4 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 2*x + 4 = 0 undefined where x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | lines differ by the constant -3*log(2)/2 undefined where x = 0 undefined where 2*x + 4 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | lines differ by the constant 3*log(2)/2 undefined where x = 0 undefined where 2*x + 4 = 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 (error) — Step 3 incorrectly rewrites (x-1)/(x*(x+2)) as 1/(x+2)-1/(x*(x+2)). The correct partial‑fraction decomposition is -1/(2x)+3/(2(x+2)), so the subsequent algebraic manipulations are based on a false identity.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-25gpt-oss:20b: fail (error) 2026-10-25 — Step 3 incorrectly rewrites (x-1)/(x*(x+2)) as 1/(x+2)-1/(x*(x+2)). The correct partial‑fraction decomposition is -1/(2x)+3/(2(x+2)), so the subsequent algebraic manipulations are based on a false identity.qwen3.6:27b-mlx: pass 2026-10-24gpt-oss:20b: fail (error) 2026-10-24 — Step 3 incorrectly rewrites (x-1)/(x*(x+2)) as 1/(x+2)+(-1)/(x*(x+2)). The correct partial‑fraction decomposition is -1/(2x)+3/(2(x+2)), so the subsequent steps are based on an incorrect algebraic manipulation.qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly applies algebraic manipulation to decompose the integrand into partial fractions and then integrates term by term. Each step applies a single rule and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-24 — Step 3 incorrectly rewrites the integrand; the correct partial‑fraction decomposition is \(\frac{x-1}{x(x+2)}=-\frac12\frac1x+\frac32\frac1{x+2}\). The subsequent algebraic manipulations are based on this wrong expression, so the solution is mathematically incorrect.
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 2026-10-25 with SymPy 1.14.0.