Answer
False
Work Step by Step
\[\begin{align}
& \int_{-\infty }^{\infty }{x{{e}^{-2x}}}dx=\underset{c\to \infty }{\mathop{\lim }}\,\int_{-c}^{c}{x{{e}^{-2x}}}dx \\
& \text{The statement is false, we must split the limits with a constant} \\
& \text{between }\left( -\infty ,\infty \right) \\
& \int_{-\infty }^{\infty }{x{{e}^{-2x}}}dx=\underset{c\to \infty }{\mathop{\lim }}\,\int_{-\infty}^{c}{x{{e}^{-2x}}}dx+\underset{c\to \infty }{\mathop{\lim }}\,\int_{c}^{\infty}{x{{e}^{-2x}}}dx \\
& \text{False} \\
\end{align}\]
