Integral of \( \displaystyle \frac{x}{\sqrt{x^{2} + 4}} \)
Problem 4.355 · medium
Find \( \displaystyle \int \frac{x}{\sqrt{x^{2} + 4}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{x}{\sqrt{x^{2} + 4}}\, dx \]integral rewrite algebraStart with the integral of the given function. Rewrite the square root using a fractional exponent. Rewrite x as 1/2 * 2x to prepare for substitution.✓ Proved
- \[ = \sqrt{x^{2} + 4} \]antiderivativeApply the power rule for integration via substitution.✓ Proved
Answer \( \sqrt{x^{2} + 4} + C \)
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.
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 x**2 + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**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: fail (error) — Step 4 applies two rules at once (substitution and the power rule). Each step must change only one thing, so the substitution should be a separate step before applying the power rule.qwen3.6:27b-mlx: pass — The solution correctly applies substitution implicitly by rewriting the integrand to match the derivative of the inner function, and the final step correctly identifies the antiderivative.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies substitution implicitly by rewriting the integrand to match the derivative of the inner function, and the final step correctly identifies the antiderivative.gpt-oss:20b: fail (error) 2026-10-05 — Step 4 applies two rules at once (substitution and the power rule). Each step must change only one thing, so the substitution should be a separate step before applying the power rule.qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the structure for u-substitution and applies the antiderivative rule. Each step changes only one aspect of the expression or applies a single logical rule.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.