Hello
I would be appreciated if you could guide me on how to integrate the product of an arbitrary function and Dirac delta function as attached?
For a quick answer, use the free online version of Wolfram Alpha:
https://www.wolframalpha.com/input/?i=integrate++%28exp%28-B*x%29+*+DiracDelta%5Bt-x%5D%29++dx+from+x%3D0+to+t
Sure that the upper limit lands right in the middle of the delta function?
A somewhat vague answer is that \int_M f(x) \delta(t-x) dx = f(t) for any measurable subset M of the real axis containing t, and any measurable (real) function f defined on M. In your case the answer would be then e^{-\beta t} - but the integral should be understood as integral over (0,t] and not (0,t), in this case it does matter. You may find some elementary explanation of the properties of Dirac delta on wikipedia: https://en.wikipedia.org/wiki/Dirac_delta_function#As_a_measure, and a more detailed treatment in any reasonable book on measure theory. Literature on the Laplace transform (where this problem probably comes from) may also be helpful if the mathematics behind the concept is treated properly there.
Precise if * means the standard multiplication \cdot so that your question is a simple convolution integral, (a standard exercise!), or if * means a convolution operator.
Work the integral with a representation of the delta function, and then take the limit of the delta function with no width. For example a Lorenzian function.
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