Integral of \( \displaystyle \frac{2 x + 1}{\sqrt{\left(2 x + 1\right)^{2} + 4}} \)
Problem 4.1298 · hard Video
Find \( \displaystyle \int \frac{2 x + 1}{\sqrt{\left(2 x + 1\right)^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x + 1}{\sqrt{\left(2 x + 1\right)^{2} + 4}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{4 x + 2}{2 \sqrt{\left(2 x + 1\right)^{2} + 4}}\, dx \]algebraRewrite the numerator to expose the derivative of the inner function.✓ Proved
- \[ = \frac{\int \frac{4 x + 2}{\sqrt{\left(2 x + 1\right)^{2} + 4}}\, dx}{2} \]linearityPull the constant factor out of the integral.✓ Proved
- \[ = \frac{\sqrt{\left(2 x + 1\right)^{2} + 4}}{2} \]antiderivativeIntegrate the remaining expression using the substitution rule for the square root.✓ Proved
Answer \( \sqrt{\left(x + \frac{1}{2}\right)^{2} + 1} + C \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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)**2 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 4 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 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 manipulation to expose the derivative of the inner function, uses linearity to pull out constants, and applies the antiderivative rule for the form f'(x)/sqrt(f(x)^2 + a^2). The final answer is algebraically equivalent to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-21 — The solution correctly applies algebraic manipulation to expose the derivative of the inner function, uses linearity to pull out constants, and applies the antiderivative rule for the form f'(x)/sqrt(f(x)^2 + a^2). The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: pass 2026-10-20 — The solution correctly applies algebraic manipulation to prepare for substitution, uses linearity to factor out constants, and identifies the antiderivative of the resulting form. The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (error) 2026-10-20 — Step 4 applies the antiderivative rule but fails to account for the chain rule factor of 2 in the numerator, resulting in an answer that is off by a factor of 2. The derivative of the stated answer is (2*x + 1)/sqrt((2*x + 1)**2 + 4), but the integral of that function is sqrt((2*x + 1)**2 + 4), not 1/2 * sqrt((2*x + 1)**2 + 4). Wait, let me re-evaluate. d/dx(sqrt(u^2+4)) = (1/(2*sqrt(u^2+4))) * 2u * u'. Here u = 2x+1, u'=2. So d/dx(sqrt((2x+1)^2+4)) = (2*(2x+1))/sqrt((2x+1)^2+4). The integrand in step 3 is 2*(2x+1)/sqrt(...). So the antiderivative of the integrand in step 3 is indeed sqrt((2x+1)^2+4). Step 4 says 1/2 * sqrt(...). This is incorrect. The factor 1/2 was pulled out in step 3. The remaining integral is Integral(2*(2x+1)/sqrt(...), x). Let u = (2x+1)^2+4. du = 2*(2x+1)*2 dx = 4*(2x+1) dx. This doesn't match directly. Let's look at step 2. Step 2 rewrites (2x+1) as 1/2 * 2*(2x+1). This is algebraically correct. Step 3 pulls out 1/2. The remaining integral is Integral(2*(2x+1)/sqrt((2x+1)^2+4), x). Let v = (2x+1)^2+4. dv/dx = 2*(2x+1)*2 = 4*(2x+1). So 2*(2x+1) dx = 1/2 dv. The integral becomes Integral(1/sqrt(v) * 1/2 dv) = 1/2 * 2*sqrt(v) = sqrt(v). So the result of the integral in step 3 is sqrt((2x+1)^2+4). Step 4 claims the result is 1/2 * sqrt((2x+1)^2+4). This is wrong. The 1/2 factor was already outside. The integral inside evaluates to sqrt(...). So the total answer should be 1/2 * sqrt(...). Wait. Let's re-read step 4. Step 4 says: 1/2 * Integral(...) -> 1/2 * sqrt(...). This implies Integral(...) = sqrt(...). Is Integral(2*(2x+1)/sqrt((2x+1)^2+4), x) equal to sqrt((2x+1)^2+4)? Derivative of sqrt((2x+1)^2+4) is (1/(2*sqrt(...))) * 2*(2x+1)*2 = 2*(2x+1)/sqrt(...). Yes. So the integral is correct. The final answer is 1/2 * sqrt((2x+1)^2+4). The stated answer is sqrt((x+1/2)^2+1). Note that (2x+1)^2+4 = 4(x+1/2)^2+4 = 4((x+1/2)^2+1). So sqrt((2x+1)^2+4) = 2*sqrt((x+1/2)^2+1). Therefore 1/2 * sqrt((2x+1)^2+4) = 1/2 * 2 * sqrt((x+1/2)^2+1) = sqrt((x+1/2)^2+1). The final answer matches. The steps are correct. Why did I think it was wrong? I misread step 4 as claiming the integral itself was 1/2*sqrt, but it claims the whole expression 1/2 * Integral is 1/2 * sqrt. This is correct. The label 'antiderivative' is acceptable for recognizing the form. The step changes one thing (the integral sign and integrand into the result). It is a valid step. Verdict: pass.gpt-oss:20b: pass 2026-10-20
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 2026-10-21 with SymPy 1.14.0.