Final answer:
The statement that \[ \int_{0}^{\infty} x e^{x} d x \] is an improper integral of Type II is true because the interval of integration includes infinity. However, this integral does not converge to a finite value.
Step-by-step explanation:
The statement "\[ \int_{0}^{\infty} x e^{x} d x \] is an improper integral of Type II" is True. An improper integral of Type II occurs when the interval of integration is infinite, such as from 0 to \( \infty \). However, the example given within the student's question appears to have a mistake, since the integral \[ \int_{0}^{\infty} x e^{x} d x \] does not converge and therefore does not result in a meaningful value.
An improper integral is evaluated using limit processes where a bound of the integral is replaced by a limit. For the integral to be valid, the limit must exist and be finite. In this case, evaluating \[ \int_{0}^{\infty} x e^{x} d x \] would involve taking the limit as the upper bound approaches infinity, which does not exist because the function \( x e^{x} \) increases without bound as \( x \) goes to infinity.