Integral of \( \displaystyle 4 x \sqrt{4 x^{2} + 1} \)
Problem 4.1261 · medium Video
Find \( \displaystyle \int 4 x \sqrt{4 x^{2} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int 4 x \sqrt{4 x^{2} + 1}\, dx \]integral algebraStart with the integral of the given function. Rewrite the integrand to facilitate substitution.✓ Proved
- \[ = \frac{\int 8 x \sqrt{4 x^{2} + 1}\, dx}{2} \]linearity rewriteMove the constant outside the integral. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = \frac{\left(4 x^{2} + 1\right)^{\frac{3}{2}}}{3} \]antiderivativeApply the power rule for integration using the substitution u = 4x^2 + 1.✓ Proved
Answer \( \frac{\left(4 x^{2} + 1\right)^{\frac{3}{2}}}{3} + C \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 (style) — Step 5 applies both a substitution and a power‑rule integration in a single step, violating the rule that each step must change only one thing. It should be split into a substitution step followed by a power‑rule step.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19gpt-oss:20b: fail (style) 2026-10-19 — Step 5 applies both a substitution and a power‑rule integration in a single step, violating the rule that each step must change only one thing. It should be split into a substitution step followed by a power‑rule step.qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies algebraic manipulation to prepare for substitution, uses linearity to extract constants, and applies the antiderivative rule via implicit substitution. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-10-17 — The final antiderivative is missing a factor of 2. The step 5 applies both a substitution (u=4x^2+1, du=8x dx) and the power‑rule integration, which violates the one‑rule‑per‑step rule. The correct result is 2/3*(4*x**2+1)**(3/2).qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 5 applies the antiderivative rule but skips the necessary substitution step (u = 4x^2 + 1) and the associated constant adjustment, violating the 'one thing per step' constraint. The transition from Step 4 to Step 5 combines substitution, power rule integration, and back-substitution into a single step.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.