Integral of \( \displaystyle \sqrt{2 x + 1} \)
Problem 4.652 · medium
Find \( \displaystyle \int \sqrt{2 x + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \sqrt{2 x + 1}\, dx \]integral algebra rewrite simplifyStart with the integral of the function. Rewrite the square root as a fractional exponent. Prepare for the power rule by expressing the derivative of the inner function. Simplify the integrand.✓ Proved
- \[ = \frac{\left(2 x + 1\right)^{\frac{3}{2}}}{3} \]antiderivative simplifyApply the power rule for integration. Simplify the final coefficient.✓ Proved
Answer \( \frac{\left(2 x + 1\right)^{\frac{3}{2}}}{3} + C \)
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. 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 |
| 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 |
| 6 | ✓ 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 (error) — Step 5 applies two rules at once: substitution (u=2x+1) and the power rule for integration. The label "antiderivative" is insufficient and the step violates the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) — Step 3 introduces factors (3/2 and 2) that are not algebraically equivalent to the integrand in Step 2, violating the requirement that consecutive lines be algebraically equal. Step 5 applies the power rule incorrectly by failing to account for the chain rule factor of 2 in the denominator properly relative to the previous step's structure.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — Step 5 applies two rules at once: substitution (u=2x+1) and the power rule for integration. The label "antiderivative" is insufficient and the step violates the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 introduces factors (3/2 and 2) that are not algebraically equivalent to the integrand in Step 2, violating the requirement that consecutive lines be algebraically equal. Step 5 applies the power rule incorrectly by failing to account for the chain rule factor of 2 in the denominator properly relative to the previous step's structure.gpt-oss:20b: fail (error) 2026-10-08 — Step 3 applies multiple transformations at once (introducing a factor of 1/3·(3/2)·2) and then step 4 immediately undoes it, violating the rule‑by‑step requirement. The manipulation is unnecessary and the labeling as "rewrite" is misleading because it effectively multiplies the integrand by 1, not a true rewrite of the integrand.qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 3 introduces factors (1/3, 3/2, 2) that are not present in the previous step, violating the single-rule constraint and creating an algebraic discontinuity. Step 5 applies the power rule incorrectly by dividing by the inner derivative (2) instead of multiplying by it (or handling the chain rule via substitution), resulting in an incorrect coefficient.
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 2026-10-08 with SymPy 1.14.0.