Integral of \( \displaystyle \left(2 x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{2 x - 1}} \)
Problem 4.1490 · hard Video
Find \( \displaystyle \int \left(2 x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{2 x - 1}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\left(2 x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{2 x - 1}}\right)\, dx \]integral rewriteStart with the integral of the given function. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = - \int \frac{1}{\sqrt{2 x - 1}}\, dx + \int \left(2 x - 1\right)^{\frac{3}{2}}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \int \left(\left(2 x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{2 x - 1}}\right)\, dx \]algebraFactor out the constant 2 from the integrand.✓ Proved
- \[ = - \int \frac{1}{\sqrt{2 x - 1}}\, dx + \int \left(2 x - 1\right)^{\frac{3}{2}}\, dx \]simplifySimplify the expression inside the integral.✓ Proved
- \[ = \frac{\left(2 x - 1\right)^{\frac{5}{2}}}{5} - \sqrt{2 x - 1} \]antiderivative simplifyIntegrate each term using the power rule and substitution. Simplify the final coefficients.✓ Proved
Answer \( \frac{\left(2 x - 1\right)^{\frac{5}{2}}}{5} - \sqrt{2 x - 1} + C \)
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. 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 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 7 | ✓ 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 4 incorrectly factors a 2 from the integrand, applying two rules at once and changing the expression. The subsequent steps rely on this incorrect manipulation, leading to an incorrect antiderivative.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 4 is mathematically incorrect; it claims to factor out a 2 from the integrand, but the resulting expression inside the integral does not equal
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-11-11 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 4 is mathematically incorrect; it claims to factor out a 2 from the integrand, but the resulting expression inside the integral does not equalgpt-oss:20b: fail (error) 2026-11-11 — Step 4 incorrectly factors a 2 from the integrand, applying two rules at once and changing the expression. The subsequent steps rely on this incorrect manipulation, leading to an incorrect antiderivative.qwen3.6:27b-mlx: fail (error) 2026-11-10 — Step 4 is mathematically incorrect; factoring out 2 from the integrand does not yield the expression shown, and Step 5 reverts to the previous state without justification. Furthermore, Step 6 applies the antiderivative rule to the wrong expression (the result of Step 5, not Step 4), making the logical flow broken.gpt-oss:20b: inconclusive 2026-11-10 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 4 incorrectly factors a 2 out of the integrand, which is not a valid algebraic manipulation of the expression. The integrand \((2x-1)^{3/2}-(2x-1)^{-qwen3.6:27b-mlx: fail (error) 2026-11-10 — Step 4 is mathematically incorrect; it claims to factor out a 2, but the expression inside the integral does not equal the original integrand (e.g., 2 * 1/2 * (2x-1)^(3/2) = (2x-1)^(3/2), but the second term becomes -1/2 * (2x-1)^(-1/2) instead of -(2x-1)^(-1/2)). Furthermore, Step 5 reverts to the previous state, making the sequence logically incoherent.gpt-oss:20b: fail (error) 2026-11-10 — Step 6 applies the antiderivative rule to the integrand that still has an extraneous factor 1/2 from step 4, resulting in incorrect coefficients. The correct antiderivative of (2x‑1)^(3/2) is (2/5)(2x‑1)^(5/2) without the extra 1/2, and similarly for the second term. This step combines two rules and introduces a computational error.
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 2026-11-11 with SymPy 1.14.0.