Answer
=1
Work Step by Step
\[\begin{gathered}
\int_0^\infty {x{e^{ - x}}dx} \hfill \\
\hfill \\
use\,\,the\,\,Formula\,\,for\,\,in\,tegration\,\,by\,\,parts \hfill \\
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\int_{}^{} {udv = uv - \int_{}^{} {vdu} } \hfill \\
\hfill \\
set \hfill \\
du = x\,\,\,\,\,\,\,\,\,\,\,\,\,\,then\,\,\,du = dx \hfill \\
dv = {e^{ - x}}dx\,\,\,then\,\,\,\,v = {e^{ - x}} \hfill \\
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\operatorname{int} egrating \hfill \\
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\int_0^\infty {x{e^{ - x}}dx} \,\, = \,\,\,\mathop {\lim }\limits_{b \to \infty } \int_0^b {x{e^{ - x}}dx} \hfill \\
\hfill \\
= \mathop {\lim }\limits_{b \to \infty } \,\,\left[ {x\,\left( { - {e^{ - x}}} \right) - {e^{ - x}}} \right]_0^b \hfill \\
\hfill \\
use\,\,the\,\,ftc \hfill \\
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= \mathop {\lim }\limits_{b \to \infty } \,\,\left[ { - b\,\left( {{e^{ - b}}} \right) - {e^{ - b}} + 1} \right] \hfill \\
\hfill \\
evaluate\,\,the\,\,{\text{limit}} \hfill \\
\hfill \\
= 1 \hfill \\
\end{gathered} \]
