Integration Basics

Integration by parts is the product rule read backwards. It is the workhorse technique for integrals involving a product of "algebraic" and "transcendental" pieces.

State the integration-by-parts formula and use it to evaluate

${\int }_0^{1}x{e}^{x}dx.$

Explain how you chose which factor to call $u$ and which to call $dv$, and why the alternative choice would not terminate.