Integral of \( \displaystyle \frac{2 x + 1}{\left(2 x + 2\right) \left(2 x + 4\right)} \)
Problem 4.803 · medium
Find \( \displaystyle \int \frac{2 x + 1}{\left(2 x + 2\right) \left(2 x + 4\right)} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x + 1}{\left(2 x + 2\right) \left(2 x + 4\right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{2 x + 1}{4 x^{2} + 12 x + 8}\, dx \]algebraExpand the denominator.✓ Proved
- \[ = \int \frac{2 x + 1}{\left(x + 2\right) \left(4 x + 4\right)}\, dx \]algebraFactor the denominator.✓ Proved
- \[ = \int \left(\frac{3}{4 x + 8} - \frac{1}{4 x + 4}\right)\, dx \]partial-fractionsDecompose the integrand using partial fractions.✓ Proved
- \[ = - \frac{\ln{\left(x + 1 \right)}}{4} + \frac{3 \ln{\left(x + 2 \right)}}{4} \]antiderivativeIntegrate each term separately.✓ Proved
Answer \( \frac{- \ln{\left(x + 1 \right)} + 3 \ln{\left(x + 2 \right)}}{4} + C \)
✓ 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.
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 + 2 = 0 undefined where 2*x + 4 = 0 undefined where 4*x**2 + 12*x + 8 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 12*x + 8 = 0 undefined where x + 2 = 0 undefined where 4*x + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 2 = 0 undefined where 4*x + 4 = 0 undefined where 4*x + 8 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 8 = 0 undefined where 4*x + 4 = 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies partial fraction decomposition and integrates term-by-term. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.