Integral of \( \displaystyle \frac{1}{\left(x - 1\right)^{2}} \)
Problem 4.884 · easy
Find \( \displaystyle \int \frac{1}{\left(x - 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(x - 1\right)^{2}}\, dx \]integral rewrite rewriteStart with the integral of the given function. Rewrite the negative exponent as a fraction. Revert to power form for integration.✓ Proved
- \[ = - \frac{1}{x - 1} \]antiderivativeApply the power rule for integration.✓ Proved
Answer \( - \frac{1}{x - 1} + C \)
✓ 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| 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 the power rule for integration via substitution (implicitly handled by the single-step antiderivative rule). The intermediate rewrite steps are valid algebraic manipulations.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the power rule for integration via substitution (implicitly handled by the single-step antiderivative rule). The intermediate rewrite steps are valid algebraic manipulations.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the power rule for integration. The intermediate rewrite steps are stylistically redundant but algebraically valid and do not violate the single-rule-per-step constraint.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.