Integral of \( \displaystyle \frac{x + 1}{\left(x + 1\right)^{2} + 1} \)
Problem 4.262 · medium
Find \( \displaystyle \int \frac{x + 1}{\left(x + 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{x + 1}{\left(x + 1\right)^{2} + 1}\, dx \]algebraExpand the denominator.✓ Proved
- \[ = \int \frac{x + 1}{x^{2} + 2 x + 2}\, dx \]rewriteRewrite the numerator to match the derivative of the denominator.✓ Proved
- \[ = \int \frac{2 x + 2}{2 x^{2} + 4 x + 4}\, dx \]antiderivativeIntegrate using the logarithmic rule.✓ Proved
- \[ = 0.5 \ln{\left(x^{2} + 2 x + 2 \right)} \]simplifySimplify the expression.✓ Proved
Answer \( \frac{\ln{\left(x^{2} + 2 x + 2 \right)}}{2} + C \)
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.
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)**2 + 1 = 0 undefined where x**2 + 2*x + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 2*x + 2 = 0 undefined where 2*x**2 + 4*x + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x**2 + 4*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 applies two transformations at once: it rewrites the integrand as “(2x+2)/(2(x^2+2x+2))” and then immediately invokes the logarithmic antiderivative rule. This violates the one‑rule‑per‑step rule and the label "antiderivative" is misleading because the step also performs a rewrite.qwen3.6:27b-mlx: fail (error) — Step 3 applies the integration rule (antiderivative) to evaluate the integral, but the label 'antiderivative' is not in the allowed vocabulary (which lists 'integral' for this operation). Additionally, Step 3 performs the integration in a single jump from the integrand to the result, skipping the explicit substitution step required by the 'one change per step' rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 3 applies the integration rule (antiderivative) to evaluate the integral, but the label 'antiderivative' is not in the allowed vocabulary (which lists 'integral' for this operation). Additionally, Step 3 performs the integration in a single jump from the integrand to the result, skipping the explicit substitution step required by the 'one change per step' rule.gpt-oss:20b: fail (error) 2026-10-04 — Step 3 applies two transformations at once: it rewrites the integrand as “(2x+2)/(2(x^2+2x+2))” and then immediately invokes the logarithmic antiderivative rule. This violates the one‑rule‑per‑step rule and the label "antiderivative" is misleading because the step also performs a rewrite.qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 1 is labeled 'algebra' but the note claims to expand the denominator, which is not performed in the step (the denominator remains factored). Step 3 is labeled 'antiderivative' but performs an algebraic rewrite of the integrand rather than evaluating the integral; the actual integration happens in this step but is mislabeled, and the transition from step 2 to 3 involves algebraic manipulation not captured by the label.gpt-oss:20b: fail (style) 2026-10-04 — Step 3 applies two changes (factoring 1/2 and rewriting the numerator) in one line and is labeled "antiderivative" even though it only rewrites the integrand. This violates the one‑rule‑per‑step rule and mislabels the operation.
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 2026-10-04 with SymPy 1.14.0.