Definite integrals by substitution
Problem 4.768 · medium
Evaluate \( \displaystyle \int_{0}^{2} x \left(x^{2} + 1\right)^{4}\, dx \).
- Let u = x**2 + 1; then du = 2*x dx, which is in the integrand up to a constant.
- \[ \left. x^{2} + 1 \right|_{\substack{ x=0 }} = 1 \]The lower limit in u.✓ Proved
- \[ \left. x^{2} + 1 \right|_{\substack{ x=2 }} = 5 \]The upper limit in u.✓ Proved
- \[ \int\limits_{1}^{5} \frac{u^{4}}{2}\, du = \frac{1562}{5} \]Integrate in u between the new limits; no back-substitution needed.✓ Proved
Answer \( \frac{1562}{5} \)
Lines: 3 proved, 1 not checked. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (error) — The final integral was evaluated incorrectly: ∫(u⁴/2)du from 1 to 5 equals (5⁵−1)/10 = 782/5, not 1562/5.qwen3.6:27b-mlx: pass — The solution correctly applies u-substitution, properly transforms the limits of integration, and accounts for the constant factor in the differential. The final arithmetic is correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The final integral was evaluated incorrectly: ∫(u⁴/2)du from 1 to 5 equals (5⁵−1)/10 = 782/5, not 1562/5.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies u-substitution, properly transforms the limits of integration, and accounts for the constant factor in the differential. The final arithmetic is correct.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the method of substitution, including the proper transformation of limits and the constant factor. The final result is correct.
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 2026-10-09 with SymPy 1.14.0.