Explanation

Given,

Function f(x) = x e^|x|

 Integral is -1 to 1.

If f(-x) = f(x) then the function is said to be even function

 If f(-x) = - f(x) then the function is said to be odd function.

 f(-x) = -x e^|-x| = -x e^|x| = - f(x)

 ∴ The given function is an odd function.

 For an odd function:

\(\mathop \smallint \nolimits_{ - a}^a x\;{f(x)}dx\) = 0

For a even function

\(\mathop \smallint \nolimits_{ - a}^a x\;{f(x)}dx\) = 2 × \(\mathop \smallint \nolimits_{ 0}^a x\;{f(x)}dx\)

 Now, as the function is odd

\(\mathop \smallint \nolimits_{ - 1}^1 x\;{e^{\left| x \right|}}dx\) = 0