Integral of \( \displaystyle \frac{2 x + 3}{x + \left(x + 1\right)^{2} + 4} \)
Problem 4.1123 · medium Video
Find \( \displaystyle \int \frac{2 x + 3}{x + \left(x + 1\right)^{2} + 4} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x + 3}{x + \left(x + 1\right)^{2} + 4}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{2 x + 3}{x^{2} + 3 x + 5}\, dx \]algebra simplify algebra simplify rewriteExpand the squared term. Combine like terms in the denominator. Rewrite the numerator to isolate the derivative of the denominator. Simplify the numerator expression. Identify the derivative of the denominator.✓ Proved
- \[ = \ln{\left(x^{2} + 3 x + 5 \right)} \]antiderivativeIntegrate using the substitution u = x**2 + 3*x + 5.✓ Proved
Answer \( \ln{\left(x^{2} + 3 x + 5 \right)} + C \)
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 + (x + 1)**2 + 4 = 0 undefined where x**2 + 3*x + 5 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 3*x + 5 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 3*x + 5 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 3*x + 5 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 3*x + 5 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 3*x + 5 = 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: fail (error) — Step 4 introduces a complex expression (adding and subtracting the denominator) but Step 5 simplifies it back to the original numerator without performing the split into two separate integrals required to isolate the derivative term. The step labeled 'simplify' in Step 5 effectively undoes the work of Step 4, making Step 4 redundant and the transition to Step 7 logically disconnected because the split necessary for the substitution method was never actually executed.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 4 introduces a complex expression (adding and subtracting the denominator) but Step 5 simplifies it back to the original numerator without performing the split into two separate integrals required to isolate the derivative term. The step labeled 'simplify' in Step 5 effectively undoes the work of Step 4, making Step 4 redundant and the transition to Step 7 logically disconnected because the split necessary for the substitution method was never actually executed.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly simplifies the denominator and identifies the numerator as the derivative of the denominator, leading to the correct logarithmic antiderivative. The steps adhere to the one-change-per-step rule and use valid labels.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies that the numerator is the derivative of the denominator and applies the standard logarithmic integration rule. All steps are algebraically valid and properly labeled.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.