Integral of \( \displaystyle 9 x^{2} \ln{\left(3 x \right)} \)
Problem 4.1124 · medium Video
Find \( \displaystyle \int 9 x^{2} \ln{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int 9 x^{2} \ln{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int 9 x^{2} \left(\ln{\left(x \right)} + \ln{\left(3 \right)}\right)\, dx \]algebraUse the logarithm product rule to split the log term.✓ Proved
- \[ = 9 \ln{\left(3 \right)} \int x^{2}\, dx + 9 \int x^{2} \ln{\left(x \right)}\, dx \]linearityDistribute the 9*x**2 and split into two integrals.✓ Proved
- \[ = 3 x^{3} \ln{\left(3 \right)} + 9 \int x^{2} \ln{\left(x \right)}\, dx \]antiderivative simplifyIntegrate the first term using the power rule. Simplify the coefficient of the first term.✓ Proved
- \[ = 3 x^{3} \ln{\left(x \right)} + 3 x^{3} \ln{\left(3 \right)} - 9 \int \frac{x^{2}}{3}\, dx \]parts algebra algebraApply integration by parts to the second integral with u=log(x) and dv=x**2 dx. Simplify the integrand inside the integral after differentiation. Distribute the 9 into the parentheses.✓ Proved
- \[ = 3 x^{3} \ln{\left(x \right)} + 3 x^{3} \ln{\left(3 \right)} - 3 \int x^{2}\, dx \]algebraSimplify the coefficient of the remaining integral.✓ Proved
- \[ = 3 x^{3} \ln{\left(x \right)} - x^{3} + 3 x^{3} \ln{\left(3 \right)} \]antiderivative simplifyIntegrate the final term using the power rule. Simplify the final expression.✓ Proved
Answer \( x^{3} \left(3 \ln{\left(3 x \right)} - 1\right) + C \)
✓ 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 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 integration by parts and algebraic simplification. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies integration by parts and algebraic simplification. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies integration by parts and logarithmic identities. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the logarithm product rule, linearity, and integration by parts. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.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.