Integrating an absolute value on exponential

Question

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.

Answer 1

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...