This might be a bit rusty but hopefully it can be brushed up.
I need to integrate
$$\int_{-\infty}^{\infty}xe^{-2\lambda \left | x \right |}dx$$
Recall:
$$\left | x \right |=\left\{\begin{matrix}
x &x\geq 0 \\
-x& x< 0
\end{matrix}\right.$$
Then,
$$\lim_{t\rightarrow \infty}\int_{-t}^{0}xe^{-2\lambda(-x)}dx+\lim_{t\rightarrow \infty}\int_{0}^{t}xe^{-2\lambda(x)}dx
$$
I would appreciate a nudge. Intuition suggest odd and even function have a role to play.
Absolute values are nasty.
Best Answer
Absolute value is even, so exponential is even. $x$ is odd. Odd times even is odd. Thus, an integral of an odd function about a symmetric interval is...