Dirac delta function integral

[Cujo]

I have no idea how to solve this Dirac delta function integral. So I would like some help please? 

[ataradov]

Is delta prime in this notation a standard Dirac function, or does the prime means something here?

If it is a plain Dirac function, then it would be exp(5*0). In general an integral from a to b of f(x)*delta(x-c) is f(c) for a < c < b, 1/2*f(c) if c == a or c == b, or 0 otherwise.

[Cujo]

Is delta prime in this notation a standard Dirac function, or does the prime means something here?

If it is a plain Dirac function, then it would be exp(5*0). In general an integral from a to b of f(x)*delta(x-c) is f(c) for a < c < b, 1/2*f(c) if c == a or c == b, or 0 otherwise.

I think its the derivative of the delta function? The question doesn't say. It just says solve the integrals with no context.

[ataradov]

I'm not sure if it apples to the Dirac function (partially because it is not really a function), but there is a thing where int(-inf, +inf) [f(x)*g'(x)dx] = -int(-inf, +inf) [f'(x)*g(x)dx].

I'm also not sure how this translates to definite integrals. But that may be at least direction for reseach.

[ejeffrey]

Huh.  I always thought the dirac delta function was non-differentiable but according to wikipedia you can define a differential distribution by integrating by parts the definition of the delta function.

https://en.wikipedia.org/wiki/Dirac_delta_function#Distributional_derivatives

If you evaluate that formula you get -5 * exp(5*t) | t=0 = -5

[Cujo]

Huh.  I always thought the dirac delta function was non-differentiable but according to wikipedia you can define a differential distribution by integrating by parts the definition of the delta function.

https://en.wikipedia.org/wiki/Dirac_delta_function#Distributional_derivatives

If you evaluate that formula you get -5 * exp(5*t) | t=0 = -5

This seems 100% correct, if you write the derivative delta function in Leibniz's notation, the dt cancels with the integral and your left with two "functions" multiplying together, hence use by parts.

Thank you 

[EEEnthusiast]

you can try to integrate by parts,
https://revisionmaths.com/advanced-level-maths-revision/pure-maths/calculus/integration-parts