Integral of \( \displaystyle \left(3 x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{3 x + 1}} \)
Problem 4.1409 · hard Video
Find \( \displaystyle \int \left(3 x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{3 x + 1}} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\left(3 x + 1\right)^{\frac{3}{2}} - \frac{1}{\sqrt{3 x + 1}}\right)\, dx \]integral rewriteStart with the integral of the given function. Rewrite the square root using a fractional exponent.✓ Proved
- \[ = - \int \frac{1}{\sqrt{3 x + 1}}\, dx + \int \left(3 x + 1\right)^{\frac{3}{2}}\, dx \]linearity algebraSplit the integral into two parts. Factor out the 3 from the base.✓ Proved
- \[ = - \int \frac{\sqrt{3}}{3 \sqrt{x + \frac{1}{3}}}\, dx + \int 3 \sqrt{3} \left(x + \frac{1}{3}\right)^{\frac{3}{2}}\, dx \]algebraDistribute the power to the factored terms.✓ Proved
- \[ = - \frac{\sqrt{3} \int \frac{1}{\sqrt{x + \frac{1}{3}}}\, dx}{3} + 3 \sqrt{3} \int \left(x + \frac{1}{3}\right)^{\frac{3}{2}}\, dx \]linearityPull the constant coefficients out of the integrals.✓ Proved
- \[ = \frac{6 \sqrt{3} \left(x + \frac{1}{3}\right)^{\frac{5}{2}}}{5} - \frac{2 \sqrt{3} \sqrt{x + \frac{1}{3}}}{3} \]antiderivative simplify rewriteApply the power rule for integration. Group the constants. Rewrite the term (x + 1/3) in terms of (3x + 1).✓ Proved
- \[ = \frac{2 \left(3 x + 1\right)^{\frac{5}{2}}}{15} - \frac{2 \sqrt{3 x + 1}}{3} \]algebra algebra algebra simplifyDistribute the exponent to the numerator and denominator. Combine the powers of 3. Simplify the exponents. Final simplification of the coefficients.✓ Proved
Answer \( \frac{2 \sqrt{3 x + 1} \left(\left(3 x + 1\right)^{2} - 5\right)}{15} + C \)
✓ 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. 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 3*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where x + 1/3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x + 1/3 = 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 13 | ✓ 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 — The solution correctly applies the rules of integration, algebraic manipulation, and simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-02 — The solution correctly applies the rules of integration, algebraic manipulation, and simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-02qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies the power rule for integration via substitution, handling the constant factors algebraically. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies the power rule for integration via substitution, handling the constant factors algebraically in subsequent steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-01
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 2026-11-02 with SymPy 1.14.0.