Definite integrals
Problem 4.56 · easy
Evaluate \( \displaystyle \int_{-2}^{3} x \left(x^{2} + 1\right)^{2} \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
- \[ \frac{d}{d x} \left(\frac{x^{6}}{6} + \frac{x^{4}}{2} + \frac{x^{2}}{2}\right) = x \left(x^{2} + 1\right)^{2} \]F(x) = x**6/6 + x**4/2 + x**2/2 is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. \frac{x^{6}}{6} + \frac{x^{4}}{2} + \frac{x^{2}}{2} \right|_{\substack{ x=-2 }} + \left. \frac{x^{6}}{6} + \frac{x^{4}}{2} + \frac{x^{2}}{2} \right|_{\substack{ x=3 }} = \frac{875}{6} \]F(b) − F(a).✓ Proved
Answer \( \frac{875}{6} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature (mpmath) gives the same value |
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_integral, checked 2026-09-26 with SymPy 1.14.0.