Integral of \( \displaystyle \frac{3 x + 1}{\sqrt{\left(3 x + 1\right)^{2} + 4}} \)
Problem 4.878 · hard
Find \( \displaystyle \int \frac{3 x + 1}{\sqrt{\left(3 x + 1\right)^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x + 1}{\sqrt{\left(3 x + 1\right)^{2} + 4}}\, dx \]integral rewrite substitution substitutionStart with the integral of the given function. Rewrite the square root using a fractional exponent. Let u = (3*x + 1)**2 + 4, then du = 6x dx, but we will use the chain rule logic directly. Identify the inner function 3*x + 1.✓ Proved
- \[ = \frac{\sqrt{\left(3 x + 1\right)^{2} + 4}}{3} \]antiderivativeIntegrate using the power rule and chain rule reverse.✓ Proved
Answer \( \frac{\sqrt{\left(3 x + 1\right)^{2} + 4}}{3} + C \)
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. A reviewer's objection to the explanation stands; see 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 (3*x + 1)**2 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*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: fail (error) — Step 3 applies two operations at once: it rewrites the integrand and attempts a substitution simultaneously, and the substitution is incorrectly described. The correct approach is a single substitution u = (3x+1)^2+4, du = 6(3x+1)dx, leading to the antiderivative sqrt(u)/3. The solution’s labeling and description are therefore incorrect.qwen3.6:27b-mlx: fail (error) — Step 3 is mathematically incoherent; it modifies the differential term to 'x/3 * 3' which does not correspond to a valid substitution step and leaves the integrand unchanged. Step 4 is also invalid as it claims to perform a substitution but results in the exact same expression as Step 2 without any transformation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 3 is mathematically incoherent; it modifies the differential term to 'x/3 * 3' which does not correspond to a valid substitution step and leaves the integrand unchanged. Step 4 is also invalid as it claims to perform a substitution but results in the exact same expression as Step 2 without any transformation.gpt-oss:20b: fail (error) 2026-10-11 — Step 3 applies two operations at once: it rewrites the integrand and attempts a substitution simultaneously, and the substitution is incorrectly described. The correct approach is a single substitution u = (3x+1)^2+4, du = 6(3x+1)dx, leading to the antiderivative sqrt(u)/3. The solution’s labeling and description are therefore incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 3 is mathematically incoherent; it attempts to introduce a substitution factor 'x/3 * 3' inside the integral argument, which is not a valid operation and does not represent the substitution u = (3x+1)^2 + 4. Step 4 reverts to the original integral without performing the substitution, making the logic broken.gpt-oss:20b: fail (error) 2026-10-11 — Step 3 applies two rules at once (introduces a factor of 1/3 and changes the differential) and incorrectly states the substitution. Step 4 repeats the same integrand without any change. The labeling of the substitution step is also misleading.
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.