Integral of \( \displaystyle \frac{1}{\left(2 x + 1\right) \left(2 x + 2\right)^{2}} \)
Problem 4.727 · medium
Find \( \displaystyle \int \frac{1}{\left(2 x + 1\right) \left(2 x + 2\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(2 x + 1\right) \left(2 x + 2\right)^{2}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(- \frac{1}{2 x + 2} - \frac{1}{\left(2 x + 2\right)^{2}} + \frac{1}{2 x + 1}\right)\, dx \]partial-fractionsDecompose the integrand using partial fraction decomposition.✓ Proved
- \[ = \int \frac{1}{2 x + 1}\, dx - \int \frac{1}{\left(2 x + 2\right)^{2}}\, dx - \int \frac{1}{2 x + 2}\, dx \]linearitySplit the integral into three separate integrals.≈ Checked numerically
- \[ = \frac{\ln{\left(2 x + 1 \right)}}{2} - \int \frac{1}{\left(2 x + 2\right)^{2}}\, dx - \int \frac{1}{2 x + 2}\, dx \]antiderivativeIntegrate the first term using substitution u = 2x + 1.✓ Proved
- \[ = \frac{\ln{\left(2 x + 1 \right)}}{2} - \frac{\ln{\left(2 x + 2 \right)}}{2} - \int \frac{1}{\left(2 x + 2\right)^{2}}\, dx \]antiderivativeIntegrate the second term using substitution u = 2x + 2.✓ Proved
- \[ = \frac{\ln{\left(2 x + 1 \right)}}{2} - \frac{\ln{\left(2 x + 2 \right)}}{2} + \frac{1}{4 x + 4} \]antiderivative simplifyIntegrate the third term using the power rule. Distribute the 2 in the denominator to simplify the expression.✓ Proved
Answer \( \frac{\ln{\left(x + \frac{1}{2} \right)}}{2} - \frac{\ln{\left(x + 1 \right)}}{2} + \frac{1}{4 x + 4} + C \)
Lines: 6 proved, 2 checked numerically. 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.
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 + 1 = 0 undefined where 2*x + 2 = 0 |
| 3 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log(x + 1/2)/2 - log(2*x + 1)/2 + log(2)/2; numeric agreement only, at 24 of 24 sampled points undefined where 2*x + 1 = 0 undefined where 2*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 1 = 0 undefined where 2*x + 2 = 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 undefined where 2*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where 4*x + 4 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x + 4 = 0 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -log(x + 1/2)/2 + log(2*x + 1)/2 - log(2)/2; numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments undefined where 4*x + 4 = 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 6 applies the antiderivative rule to 1/(2*x+2)**2 but produces the wrong sign. The integral of (2x+2)^{-2} is -1/(2*(2x+2)), not +1/(2*(2x+2)).qwen3.6:27b-mlx: pass — The solution correctly applies partial fraction decomposition, linearity, and antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 6 applies the antiderivative rule to 1/(2*x+2)**2 but produces the wrong sign. The integral of (2x+2)^{-2} is -1/(2*(2x+2)), not +1/(2*(2x+2)).qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies partial fraction decomposition, linearity, and antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-09 — Step 6 applies the power rule to ∫(2x+2)^{-2}dx but yields the wrong sign. The correct antiderivative is -1/(2(2x+2)), not +1/(2(2x+2)).qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies partial fraction decomposition, linearity of integration, and standard antiderivative rules. Each step modifies only one part of the expression or applies a single rule, and the labels are appropriate.
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 2026-10-09 with SymPy 1.14.0.