Integral of \( \displaystyle \frac{4 x + 1}{4 x^{2} - 1} \)
Problem 4.1491 · medium Video
Find \( \displaystyle \int \frac{4 x + 1}{4 x^{2} - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{4 x + 1}{4 x^{2} - 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{4 x + 1}{\left(2 x - 1\right) \left(2 x + 1\right)}\, dx \]algebra algebraFactor the denominator using the difference of squares. Rewrite the fraction as a product.✓ Proved
- \[ = \int \left(\frac{1}{2 \left(2 x + 1\right)} + \frac{3}{2 \left(2 x - 1\right)}\right)\, dx \]partial-fractionsApply partial fraction decomposition.✓ Proved
- \[ = \int \left(\frac{1}{4 x + 2} + \frac{3}{4 x - 2}\right)\, dx \]algebraDistribute the 1/2 into the terms.✓ Proved
- \[ = \int \left(\frac{1}{2 \left(2 x + 1\right)} + \frac{3}{2 \left(2 x - 1\right)}\right)\, dx \]algebraRewrite the division for clarity.✓ Proved
- \[ = \int \frac{3}{4 x - 2}\, dx + \int \frac{1}{4 x + 2}\, dx \]linearitySplit the integral into two parts.≈ Checked numerically
- \[ = \frac{3 \int \frac{1}{2 x - 1}\, dx}{2} + \frac{\int \frac{1}{2 x + 1}\, dx}{2} \]algebraFactor out the constants from each integral.✓ Proved
- \[ = \frac{3 \ln{\left(2 x - 1 \right)}}{4} + \frac{\ln{\left(2 x + 1 \right)}}{4} \]antiderivative simplifyIntegrate each term using the substitution rule for log(ax+b). Simplify the coefficients.✓ Proved
Answer \( \frac{3 \ln{\left(x - \frac{1}{2} \right)} + \ln{\left(x + \frac{1}{2} \right)}}{4} + C \)
✓ Nihil obstat Lines: 9 proved, 2 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 4*x**2 - 1 = 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 undefined where 4*x - 2 = 0 undefined where 4*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x - 2 = 0 undefined where 4*x + 2 = 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 |
| 7 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*log(x - 1/2)/4 + log(x + 1/2)/4 - 3*log(4*x - 2)/4 - log(4*x + 2)/4; numeric agreement only, at 24 of 24 sampled points undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 undefined where 4*x - 2 = 0 undefined where 4*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | lines differ by the constant log(2) undefined where 4*x - 2 = 0 undefined where 4*x + 2 = 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where 2*x + 1 = 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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left -3*log(x - 1/2)/4 - log(x + 1/2)/4 + 3*log(2*x - 1)/4 + log(2*x + 1)/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 3 applies two algebraic manipulations (factoring the denominator and rewriting the fraction as a product) in a single step, violating the one-rule-per-step constraint. Step 8 is labeled 'algebra' but performs the extraction of constants from integrals, which is a property of linearity; while 'algebra' is acceptable if interpreted broadly, the label 'linearity' would be more precise for the operation performed on the integral operator.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-11-11 — Step 3 applies two algebraic manipulations (factoring the denominator and rewriting the fraction as a product) in a single step, violating the one-rule-per-step constraint. Step 8 is labeled 'algebra' but performs the extraction of constants from integrals, which is a property of linearity; while 'algebra' is acceptable if interpreted broadly, the label 'linearity' would be more precise for the operation performed on the integral operator.gpt-oss:20b: pass 2026-11-11qwen3.6:27b-mlx: pass 2026-11-10 — The solution correctly applies partial fraction decomposition and integration rules. The final answer is algebraically equivalent to the stated answer, differing only by the absorption of constant terms into the constant of integration.gpt-oss:20b: pass 2026-11-10qwen3.6:27b-mlx: fail (style) 2026-11-10 — Step 3 is redundant and confusing; it rewrites a single fraction as a product of fractions without changing the structure meaningfully, violating the principle that each step should change one thing meaningfully. Step 4 correctly applies partial fractions, making Step 3 unnecessary clutter.gpt-oss:20b: pass 2026-11-10
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 2026-11-11 with SymPy 1.14.0.