# Thread: integration of dirac delta times exponential

1. ## integration of dirac delta times exponential

I have some confusion with this integral
$\int$exp(-k(2t-t$^{,}$-t$^{,,}$))$\delta
$
(t$^{,}$-t$^{,,}$)dt$^{,,}$= exp(-2k(t-t$^{,}$))

where exp(...) is the exponential function, the integration from 0 to t, k is a constant (and also t$^{,}$), $\delta$(t$^{,}$-t$^{,,}$) is Dirac delta function which equals 1 when t$^{,}$=t$^{,,}$ and zero when t$^{,}$≠t$^{,,}$

can any one explain to me how this was done?

2. I notice that in this integration what happened only the variable t$^{,,}$
changed to t$^{,}$ in the exponential function, is it a rule or something in this type of integration?

3. Originally Posted by omeganeu
I notice that in this integration what happened only the variable t$^{,,}$
changed to t$^{,}$ in the exponential function, is it a rule or something in this type of integration?
The sifting property of the Dirac Delta function has been used: $\int_{- \infty}^{+\infty} \! f(x) \, \delta (x - x') \, dx = f(x')$.

4. Thank you