Integral of \( \displaystyle \frac{2 x}{\sqrt{4 x^{2} + 4}} \)
Problem 4.272 · medium
Find \( \displaystyle \int \frac{2 x}{\sqrt{4 x^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{2 x}{\sqrt{4 x^{2} + 4}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{x}{\sqrt{x^{2} + 1}}\, dx \]simplify simplify substitutionFactor out 4 from the square root to simplify the expression. Cancel the common factor of 2 in the numerator and denominator. Let u = x**2 + 1, then du = 2x dx, so x dx = 1/2 du. However, to follow the rule of not introducing new variables, we use the substitution logic directly on the integrand.✓ Proved
- \[ = \sqrt{x^{2} + 1} \]antiderivativeThe antiderivative of x/sqrt(x**2 + 1) is sqrt(x**2 + 1).✓ Proved
Answer \( \sqrt{x^{2} + 1} + C \)
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. A reviewer's objection to the explanation stands; see 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 4*x**2 + 4 = 0 undefined where x**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 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: fail (error) — Step 4 incorrectly rewrites the integrand as exp(log(x))/sqrt(x**2+1) and claims a substitution, but this changes the expression and does not follow a single allowed rule. The correct step would be to recognize that 2x dx is the derivative of x**2+1 and apply the substitution directly without introducing exp(log(x)).qwen3.6:27b-mlx: fail (error) — Step 4 is mathematically incorrect: it replaces x with exp(log(x)), which is not equivalent to x for all real x (specifically negative x), and the note describes a u-substitution that is not reflected in the algebraic transformation shown. Step 4 applies an invalid rewrite.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 4 is mathematically incorrect: it replaces x with exp(log(x)), which is not equivalent to x for all real x (specifically negative x), and the note describes a u-substitution that is not reflected in the algebraic transformation shown. Step 4 applies an invalid rewrite.gpt-oss:20b: fail (error) 2026-10-04 — Step 4 incorrectly rewrites the integrand as exp(log(x))/sqrt(x**2+1) and claims a substitution, but this changes the expression and does not follow a single allowed rule. The correct step would be to recognize that 2x dx is the derivative of x**2+1 and apply the substitution directly without introducing exp(log(x)).qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 4 is mathematically incorrect and mislabeled. It replaces 'x' with 'exp(log(x))' but labels the step 'substitution' with a note describing a u-substitution (u = x^2 + 1) that was not actually performed in the expression. The transition from Step 3 to Step 5 requires a valid substitution or recognition of the derivative form, which Step 4 fails to represent correctly.gpt-oss:20b: fail (error) 2026-10-04 — Step 4 incorrectly rewrites the integrand as exp(log(x))/sqrt(x**2+1) and claims a substitution, but no substitution is performed and the step changes more than one element at once. The labeling and justification are incorrect.
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 2026-10-04 with SymPy 1.14.0.