Integral of \( \displaystyle \left(x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{x - 1}} \)
Problem 4.232 · medium
Find \( \displaystyle \int \left(x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{x - 1}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\left(x - 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{x - 1}}\right)\, dx \]integralStart with the integral of the given expression.✓ Proved
- \[ = - \int \frac{1}{\sqrt{x - 1}}\, dx + \int \left(x - 1\right)^{\frac{3}{2}}\, dx \]linearity rewriteSplit the integral into two parts. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = \frac{2 \left(x - 1\right)^{\frac{5}{2}}}{5} - \int \frac{1}{\sqrt{x - 1}}\, dx \]antiderivativeIntegrate the first term using the power rule.✓ Proved
- \[ = \frac{2 \left(x - 1\right)^{\frac{5}{2}}}{5} - 2 \sqrt{x - 1} \]antiderivativeIntegrate the second term using the power rule.✓ Proved
Answer \( \frac{2 \sqrt{x - 1} \left(\left(x - 1\right)^{2} - 5\right)}{5} + C \)
✓ Nihil obstat 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. 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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 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 — The solution correctly applies linearity, rewrite, and antiderivative rules in separate steps. The final answer is algebraically equivalent to the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies linearity, rewrite, and antiderivative rules in separate steps. The final answer is algebraically equivalent to the stated answer.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies linearity, rewrite, and antiderivative rules in separate steps. Each step changes only one aspect of the expression, and the labels are accurate.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.