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For a function $g(t),$ it is given that $\int_{- \infty}^{ + \infty} g(t)e^{-j\omega t}\:dt = \omega e^{-2\omega^{2}}$ for any real value $\omega.$ If $y(t) = \int_{- \infty}^{t}\:g(\tau)\:d\tau,$ then $\int_{- \infty}^{ + \infty}y(t)dt$ is

1. $0$
2. $-j$
3. $-\frac{j}{2}$
4. $\frac{j}{2}$