Why do I get 0 in the following integral?

The solution to the following integral should be $-2$
$$
int_{mathbb{R}} frac{1}{2pi}expleft{-frac{v^2}{2(1 – u^2)}right} frac{|v|}{(1 – u^2)^{frac{3}{2}}} dv = -2
$$

However, I obtain $0$ (this looks to me just like the integral of $int_{-infty}^{infty} e^{-x^2}xdx$).
$$
– frac{1}{2pi sqrt{1 – u^2}}left(lim_{t to-infty} int_{t}^0 – frac{|v|}{(1 – u^2)}expleft{-frac{v^2}{2(1 – u^2)}right}dv + lim_{ttoinfty} int_{0}^{t} – frac{|v|}{(1 – u^2)}expleft{-frac{v^2}{2(1 – u^2)}right}dvright) = 0
$$

What am I doing wrong? This integral appeared here.