Definite integrals
Problem 4.8 · easy
Evaluate \( \displaystyle \int_{0}^{3} - 3 x^{3} - 4 x^{2} - 3 x + 1 \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
- \[ \frac{d}{d x} \left(- \frac{3 x^{4}}{4} - \frac{4 x^{3}}{3} - \frac{3 x^{2}}{2} + x\right) = - 3 x^{3} - 4 x^{2} - 3 x + 1 \]F(x) = -3*x**4/4 - 4*x**3/3 - 3*x**2/2 + x is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. - \frac{3 x^{4}}{4} - \frac{4 x^{3}}{3} - \frac{3 x^{2}}{2} + x \right|_{\substack{ x=0 }} + \left. - \frac{3 x^{4}}{4} - \frac{4 x^{3}}{3} - \frac{3 x^{2}}{2} + x \right|_{\substack{ x=3 }} = - \frac{429}{4} \]F(b) − F(a).✓ Proved
Answer \( - \frac{429}{4} \)
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.