Integral of \( \displaystyle \left(2 x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{2 x + 1}} \)
Problem 4.1318 · 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.)
- \[ \int \left(\left(2 x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{2 x + 1}}\right)\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = - \int \frac{1}{\sqrt{2 x + 1}}\, dx + \int \left(2 x + 1\right)^{\frac{3}{2}}\, dx \]linearity rewriteSplit the integral into two parts. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = - \sqrt{2 x + 1} + \int \left(2 x + 1\right)^{\frac{3}{2}}\, dx \]substitution simplifyIntegrate the second term using substitution. Simplify the coefficient of the second term.✓ Proved
- \[ = \frac{\left(2 x + 1\right)^{\frac{5}{2}}}{5} - \sqrt{2 x + 1} \]substitution simplifyIntegrate the first term using substitution. Simplify the final expression.✓ 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. 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 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 applies the substitution rule incorrectly: ∫(2x+1)^{-1/2}dx should equal 2√(2x+1), not (2√(2x+1))/2. The coefficient is wrong, leading to an incorrect antiderivative.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-23gpt-oss:20b: fail (error) 2026-10-23 — Step 4 applies the substitution rule incorrectly: ∫(2x+1)^{-1/2}dx should equal 2√(2x+1), not (2√(2x+1))/2. The coefficient is wrong, leading to an incorrect antiderivative.qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies linearity, rewrite, substitution, and simplification rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies linearity, rewrite, substitution, and simplify rules in a logical sequence. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-10-22 — Step 4 applies both substitution (u=2x+1) and integration in one line, violating the rule that each step must change only one thing. Step 6 similarly combines substitution and integration. Each of these should be split into two separate steps.
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 2026-10-23 with SymPy 1.14.0.