Integral of \( \displaystyle \frac{1}{\left(2 x + 1\right)^{2}} \)
Problem 4.1212 · medium Video
Find \( \displaystyle \int \frac{1}{\left(2 x + 1\right)^{2}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{1}{\left(2 x + 1\right)^{2}}\, dx \]integral algebra algebraStart with the integral of the given function. Rewrite the negative exponent as a reciprocal. Factor out the coefficient of x from the base.✓ Proved
- \[ = \int \frac{1}{4 \left(x + \frac{1}{2}\right)^{2}}\, dx \]algebraDistribute the square to the constant factor.✓ Proved
- \[ = \frac{\int \frac{1}{\left(x + \frac{1}{2}\right)^{2}}\, dx}{4} \]linearityPull the constant out of the integral.✓ Proved
- \[ = - \frac{1}{4 \left(x + \frac{1}{2}\right)} \]antiderivativeApply the power rule for integration.✓ Proved
- \[ = - \frac{1}{4 x + 2} \]algebra algebraSimplify the expression. Multiply the denominator by 2 to clear the fraction inside.✓ Proved
Answer \( - \frac{1}{4 x + 2} + C \)
✓ 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 undefined where 2*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 x + 1/2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/2 = 0 undefined where 4*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 4*x + 2 = 0 |
| 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 algebraic manipulations and the power rule for integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies algebraic manipulations and the power rule for integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies algebraic manipulations and the power rule for integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies algebraic manipulations and the power rule for integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.