Integral of \( \displaystyle \sqrt{3 x - 1} \)
Problem 4.1286 · medium Video
Find \( \displaystyle \int \sqrt{3 x - 1} \, dx \). (Omit the constant of integration.)
- \[ \int \sqrt{3 x - 1}\, dx \]integral algebra algebraStart with the integral of the given function. Rewrite the square root as a fractional exponent. Factor out the 3 from the expression inside the parenthesis.✓ Proved
- \[ = \int \sqrt{3} \sqrt{x - \frac{1}{3}}\, dx \]linearityPull the constant factor out of the integral.✓ Proved
- \[ = \sqrt{3} \int \sqrt{x - \frac{1}{3}}\, dx \]linearityApply the linearity rule for constants.✓ Proved
- \[ = \frac{2 \sqrt{3} \left(x - \frac{1}{3}\right)^{\frac{3}{2}}}{3} \]antiderivative algebra algebra simplifyApply the power rule for integration. Simplify the fraction resulting from the power rule. Rearrange the terms for clarity. Simplify the constant coefficient.✓ Proved
Answer \( \frac{2 \left(3 x - 1\right)^{\frac{3}{2}}}{9} + C \)
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 4 is labeled 'linearity' but performs an algebraic expansion of the integrand (splitting a product of powers) rather than applying linearity of the integral operator. Step 5 is labeled 'linearity' but merely moves the constant factor outside the integral sign, which was already done in the previous step's notation; the actual operation in step 4 was algebraic, not linear.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — Step 4 is labeled 'linearity' but performs an algebraic expansion of the integrand (splitting a product of powers) rather than applying linearity of the integral operator. Step 5 is labeled 'linearity' but merely moves the constant factor outside the integral sign, which was already done in the previous step's notation; the actual operation in step 4 was algebraic, not linear.gpt-oss:20b: pass 2026-10-20qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 4 is labeled 'linearity' but performs an algebraic expansion of the integrand (splitting a product of powers), not a linear operation on the integral operator. The constant factor is not actually pulled out until Step 5.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 4 is labeled 'linearity' but performs an algebraic expansion of the integrand (splitting (3A)^(1/2) into 3^(1/2)A^(1/2)) rather than applying the linearity of the integral operator. The actual application of linearity occurs in Step 5. This violates the single-rule-per-step constraint and mislabels the operation.gpt-oss:20b: pass 2026-10-19
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 2026-10-20 with SymPy 1.14.0.