Definite integrals
Problem 4.1 · medium
Evaluate \( \displaystyle \int_{-1}^{2} x + e^{2 x} \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
- \[ \frac{d}{d x} \left(\frac{x^{2}}{2} + \frac{e^{2 x}}{2}\right) = x + e^{2 x} \]F(x) = x**2/2 + exp(2*x)/2 is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. \frac{x^{2}}{2} + \frac{e^{2 x}}{2} \right|_{\substack{ x=-1 }} + \left. \frac{x^{2}}{2} + \frac{e^{2 x}}{2} \right|_{\substack{ x=2 }} = \frac{-1 + \left(3 + e^{4}\right) e^{2}}{2 e^{2}} \]F(b) − F(a).✓ Proved
Answer \( \frac{-1 + \left(3 + e^{4}\right) e^{2}}{2 e^{2}} \)
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.