Integral of \( \displaystyle \frac{\left(x + 1\right)^{2} + 1}{x \left(x + 1\right)} \)
Problem 4.229 · medium
Find \( \displaystyle \int \frac{\left(x + 1\right)^{2} + 1}{x \left(x + 1\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\frac{x + 1}{x} + \frac{1}{x \left(x + 1\right)}\right)\, dx \]algebra simplifySplit the fraction into two parts. Simplify the first term.✓ Proved
- \[ = \int \left(1 + \frac{1}{x} + \frac{1}{x \left(x + 1\right)}\right)\, dx \]algebraSplit the first term further.✓ Proved
- \[ = \int 1\, dx + \int \frac{1}{x}\, dx + \int \frac{1}{x \left(x + 1\right)}\, dx \]linearityApply linearity to the integral.✓ Proved
- \[ = \int 1\, dx + \int \frac{1}{x}\, dx + \int \left(- \frac{1}{x + 1} + \frac{1}{x}\right)\, dx \]partial-fractionsUse partial fraction decomposition on the third term.✓ Proved
- \[ = \int 1\, dx + 2 \int \frac{1}{x}\, dx - \int \frac{1}{x + 1}\, dx \]linearity simplifyDistribute the integral. Combine the like terms.✓ Proved
- \[ = x + 2 \ln{\left(x \right)} - \ln{\left(x + 1 \right)} \]antiderivativeIntegrate each term.✓ Proved
Answer \( x + 2 \ln{\left(x \right)} - \ln{\left(x + 1 \right)} + C \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 + 1 = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1 = 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: passqwen3.6:27b-mlx: pass — The solution correctly applies algebraic simplification, partial fraction decomposition, and linearity of integration. 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 2026-10-03 — The solution correctly applies algebraic simplification, partial fraction decomposition, and linearity of integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-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.