Integral of \( \displaystyle \frac{2 x}{\left(2 x + 1\right) \left(2 x + 3\right)} \)
Problem 4.1322 · medium Video
Find \( \displaystyle \int \frac{2 x}{\left(2 x + 1\right) \left(2 x + 3\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x}{\left(2 x + 1\right) \left(2 x + 3\right)}\, dx \]integral algebraStart with the integral of the given function. Rewrite the numerator to facilitate partial fraction decomposition.✓ Proved
- \[ = \int \left(\frac{1}{2 x + 3} - \frac{1}{\left(2 x + 1\right) \left(2 x + 3\right)}\right)\, dx \]algebraSplit the fraction into two terms.✓ Proved
- \[ = - \int \frac{1}{\left(2 x + 1\right) \left(2 x + 3\right)}\, dx + \int \frac{1}{2 x + 3}\, dx \]linearityApply the linearity property of integrals.≈ Checked numerically
- \[ = \int \frac{1}{2 x + 3}\, dx - \int \left(- \frac{1}{2 \left(2 x + 3\right)} + \frac{1}{2 \left(2 x + 1\right)}\right)\, dx \]partial-fractionsPerform partial fraction decomposition on the second term.✓ Proved
- \[ = - \frac{\int \frac{1}{2 x + 1}\, dx}{2} + \frac{3 \int \frac{1}{2 x + 3}\, dx}{2} \]linearity simplifyDistribute the constant and split the integral. Combine the like terms.≈ Checked numerically
- \[ = - \frac{\ln{\left(2 x + 1 \right)}}{4} + \frac{3 \ln{\left(2 x + 3 \right)}}{4} \]antiderivative simplifyIntegrate each logarithmic term using the substitution rule for linear functions. Multiply the constants through the expression.✓ Proved
Answer \( \frac{- \ln{\left(x + \frac{1}{2} \right)} + 3 \ln{\left(x + \frac{3}{2} \right)}}{4} + C \)
✓ Nihil obstat Lines: 7 proved, 3 checked numerically. 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 2*x + 3 = 0 undefined where 2*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 3 = 0 undefined where 2*x + 1 = 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x + 3/2)/2 - log(2*x + 3)/2; numeric agreement only, at 24 of 24 sampled points undefined where 2*x + 3 = 0 undefined where 2*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 3 = 0 undefined where 2*x + 1 = 0 |
| 6 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -log(x + 1/2)/4 + log(x + 3/2)/4 + log(2*x + 1)/4 - log(2*x + 3)/4; numeric agreement only, at 24 of 24 sampled points undefined where 2*x + 3 = 0 undefined where 2*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 3 = 0 undefined where 2*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 3 = 0 undefined where 2*x + 1 = 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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left log(x + 1/2)/4 - 3*log(x + 3/2)/4 - log(2*x + 1)/4 + 3*log(2*x + 3)/4; numeric agreement only, at 24 of 24 sampled points 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 (style) — Step 2 applies two algebraic manipulations at once: rewriting the numerator (2x = 2x+1-1) and splitting the fraction into two terms. This violates the 'one change per step' rule; the split should be a separate step.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-23 — Step 2 applies two algebraic manipulations at once: rewriting the numerator (2x = 2x+1-1) and splitting the fraction into two terms. This violates the 'one change per step' rule; the split should be a separate step.gpt-oss:20b: pass 2026-10-23qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies algebraic manipulation, partial fractions, linearity, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-22gpt-oss:20b: pass 2026-10-22
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 2026-10-23 with SymPy 1.14.0.