$\int\limits_0^1 {x{e^x}dx} = 1$

: Let $I = \int\limits_0^1 {x{e^x}dx}$

Integrating by parts, $I = \left[ {x\int {{e^x}dx} - \int {\left( {\cfrac{d}{{dx}}\left( x \right)\int {{e^x}dx} } \right)dx} } \right]_0^1$

$= \left[ {x{e^x} - \int {{e^x}dx} } \right]_0^1 = \left[ {x{e^x} - {e^x}} \right]_0^1 = \left[ {\left( {e - 0} \right) - \left( {e - 1} \right)} \right] = 1$