Explanation :
To Find value of $ \int\limits_{-\infty}^{\;\infty}\mathrm e^{-\mathrm t}\mathrm\delta\left(2\mathrm t-2\right)\mathrm{dt} $
Since $ \;\;\;\;\;\;\;\mathrm\delta\left(2\mathrm t-2\right)=\frac12\mathrm\delta(\mathrm t-1) $
above integral can be written as
$ \int\limits_{-\infty}^{\;\infty}\mathrm e^{-\mathrm t}\frac12\mathrm\delta\left(\mathrm t-1\right)\mathrm{dt}=\frac12\mathrm e^{-\mathrm t}=\frac1{2\mathrm e} $