Integral of \( \displaystyle \frac{x + 1}{\sqrt{\left(x + 1\right)^{2} + 4}} \)
Problem 4.877 · medium
Find \( \displaystyle \int \frac{x + 1}{\sqrt{\left(x + 1\right)^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{x + 1}{\sqrt{\left(x + 1\right)^{2} + 4}}\, dx \]integral rewrite algebra linearityStart with the integral of the given function. Rewrite the square root using a fractional exponent. Rewrite the numerator to prepare for substitution. Factor out the constant 1/2.✓ Proved
- \[ = \sqrt{\left(x + 1\right)^{2} + 4} \]antiderivativeThe integral follows the form u' * u^(-1/2) which integrates to 2*u^(1/2) * (1/2).✓ Proved
Answer \( \sqrt{\left(x + 1\right)^{2} + 4} + C \)
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 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 4 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 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: fail (style) — Step 4 applies the linearity rule to factor out the constant 1/2, but the integral sign is removed in the transition to Step 5 without an explicit step showing the factored form (e.g., 1/2 * Integral(...)). This violates the 'one thing per step' constraint by combining the factoring and the integration into a single jump.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-11 — Step 4 applies the linearity rule to factor out the constant 1/2, but the integral sign is removed in the transition to Step 5 without an explicit step showing the factored form (e.g., 1/2 * Integral(...)). This violates the 'one thing per step' constraint by combining the factoring and the integration into a single jump.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies substitution via algebraic manipulation and linearity, with accurate labeling and notes.gpt-oss:20b: fail (error) 2026-10-11 — The final antiderivative omits the factor 2 that arises from the substitution u=(x+1)^2+4, so the result should be 2*sqrt((x+1)^2+4), not sqrt((x+1)^2+4).
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.