Home›Calculus 1›Definite integrals by substitution›Problem 4.307

Definite integrals by substitution

Problem 4.307 · medium

Evaluate \( \displaystyle \int_{0}^{2} \frac{5 x^{2}}{\sqrt{x^{3} + 4}}\, dx \).

Let u = x**3 + 4; then du = 3*x**2 dx, which is in the integrand up to a constant.
Reviewed
\[ \left. x^{3} + 4 \right|_{\substack{ x=0 }} = 4 \]
The lower limit in u.✓ Proved
\[ \left. x^{3} + 4 \right|_{\substack{ x=2 }} = 12 \]
The upper limit in u.✓ Proved
\[ \int\limits_{4}^{12} \frac{5}{3 \sqrt{u}}\, du = - \frac{20}{3} + \frac{20 \sqrt{3}}{3} \]
Integrate in u between the new limits; no back-substitution needed.✓ Proved

Answer \( - \frac{20}{3} + \frac{20 \sqrt{3}}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature of the original integral

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies u-substitution, accurately transforms the limits of integration, and computes the definite integral correctly. The logic is sound and complete.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies u-substitution, accurately transforms the limits of integration, and computes the definite integral correctly. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the substitution method, including the proper transformation of limits and the constant factor adjustment. The final result is correct.
gpt-oss:20b: pass 2026-10-04

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 generator:structured/definite_substitution, checked 2026-10-04 with SymPy 1.14.0.