Integral of \( \displaystyle \frac{\ln{\left(3 x + 1 \right)}}{\left(3 x + 1\right)^{2}} \)
Problem 4.1400 · medium Video
Find \( \displaystyle \int \frac{\ln{\left(3 x + 1 \right)}}{\left(3 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\ln{\left(3 x + 1 \right)}}{\left(3 x + 1\right)^{2}}\, dx \]integral algebraStart with the integral of the given function. Rewrite the denominator using a negative exponent.✓ Proved
- \[ = - \int \left(- \frac{1}{\left(3 x + 1\right)^{2}}\right)\, dx - \frac{\ln{\left(3 x + 1 \right)}}{3 \left(3 x + 1\right)} \]parts algebra algebraUse integration by parts with u = log(3*x + 1) and dv = (3*x + 1)**(-2) dx. Simplify the integrand in the remaining integral. Simplify the constants in the integrand.✓ Proved
- \[ = \int \frac{1}{\left(3 x + 1\right)^{2}}\, dx - \frac{\ln{\left(3 x + 1 \right)}}{3 \left(3 x + 1\right)} \]algebraDistribute the negative sign.✓ Proved
- \[ = - \frac{\ln{\left(3 x + 1 \right)}}{3 \left(3 x + 1\right)} - \frac{1}{3 \left(3 x + 1\right)} \]antiderivativeEvaluate the integral of (3*x + 1)**(-2).✓ Proved
- \[ = - \frac{\ln{\left(3 x + 1 \right)}}{9 x + 3} - \frac{1}{9 x + 3} \]simplifyCombine the terms into a single expression.✓ Proved
Answer \( \frac{- \ln{\left(3 x + 1 \right)} - 1}{3 \left(3 x + 1\right)} + C \)
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 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 undefined where 9*x + 3 = 0 |
| 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 undefined where 9*x + 3 = 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 (error) — Step 3 applies integration by parts but fails to compute the antiderivative of dv correctly; the antiderivative of (3x+1)^(-2) is -(3x+1)^(-1)/3, not -(3x+1)^(-1). This arithmetic error propagates through the solution, resulting in an incorrect final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-31 — Step 3 applies integration by parts but fails to compute the antiderivative of dv correctly; the antiderivative of (3x+1)^(-2) is -(3x+1)^(-1)/3, not -(3x+1)^(-1). This arithmetic error propagates through the solution, resulting in an incorrect final answer.gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly applies integration by parts and simplifies the resulting expression. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly applies integration by parts and simplifies the resulting expression. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-30
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 2026-10-31 with SymPy 1.14.0.