Integral of \( \displaystyle \frac{3 x + 1}{\left(3 x + 1\right)^{2} + 1} \)
Problem 4.1265 · medium Video
Find \( \displaystyle \int \frac{3 x + 1}{\left(3 x + 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{3 x + 1}{\left(3 x + 1\right)^{2} + 1}\, dx \]integral algebraStart with the integral of the given function. Rewrite the numerator to involve the derivative of the denominator.✓ Proved
- \[ = \frac{\int \frac{18 x + 6}{\left(3 x + 1\right)^{2} + 1}\, dx}{6} \]linearityPull the constant factor out of the integral.✓ Proved
- \[ = \frac{\ln{\left(\left(3 x + 1\right)^{2} + 1 \right)}}{6} \]antiderivativeIntegrate using the logarithmic rule for the derivative of the denominator.✓ Proved
Answer \( \frac{\ln{\left(9 x^{2} + 6 x + 2 \right)}}{6} + C \)
✓ 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. 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 (3*x + 1)**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 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: pass — The solution correctly applies substitution logic via algebraic manipulation and linearity, followed by the standard logarithmic antiderivative rule. Each step changes only one aspect of the expression and uses valid labels.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies substitution logic via algebraic manipulation and linearity, followed by the standard logarithmic antiderivative rule. Each step changes only one aspect of the expression and uses valid labels.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the substitution structure, factors out the constant, and applies the logarithmic integration rule. Each step changes only one aspect of the expression and uses valid labels.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the substitution structure and applies the rules in a logical sequence. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.