Integral of \( \displaystyle \sqrt{2} \sqrt{x} \)
Problem 4.469 · medium
Find \( \displaystyle \int \sqrt{2} \sqrt{x} \, dx \). (Omit the constant of integration.)
- \[ \int \sqrt{2} \sqrt{x}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \sqrt{2} \int x^{0.5}\, dx \]linearityPull the constant out of the integral.≈ Checked numerically
- \[ = \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} \]antiderivative simplifyApply the power rule for integration. Simplify the fraction and the coefficients.≈ Checked numerically
Answer \( \frac{2 \sqrt{2} x^{\frac{3}{2}}}{3} + C \)
✓ Nihil obstat Lines: 3 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 0.666666666666667*sqrt(2)*(x**(3/2) - x**1.5); numeric agreement only, at 24 of 24 sampled points |
| 3 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 0.666666666666667*sqrt(2)*(-x**(3/2) + x**1.5); numeric agreement only, at 24 of 24 sampled points |
| 4 | ✓ 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: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.