Integral of \( \displaystyle \left(x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{x + 1}} \)
Problem 4.557 · medium
Find \( \displaystyle \int \left(x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{x + 1}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\left(x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{x + 1}}\right)\, dx \]integral rewriteStart with the integral of the given function. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = - \int \frac{1}{\sqrt{x + 1}}\, dx + \int \left(x + 1\right)^{\frac{3}{2}}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{2 \left(x + 1\right)^{\frac{5}{2}}}{5} - 2 \sqrt{x + 1} \]antiderivativeIntegrate each 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 |
| 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 the rules of integration, rewriting, linearity, and antiderivatives in distinct steps. The final answer matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the rules of integration, rewriting, linearity, and antiderivatives in distinct steps. The final answer matches the stated answer.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the rules for rewriting, linearity, and antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
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 2026-10-07 with SymPy 1.14.0.