(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

∫x*delta((x/y)-t) dx from 0 to infinity

2. Relevant equations

∫x*delta((x/y)-t) dx from 0 to infinity = ty*|y|*θ(ty)

3. The attempt at a solution

Okay, so using the transformation of variables technique via the Jacobian, I see where the |y| comes from. However, using the dirac delta method I have NO clue how that |y| is derived logically. I know that the property of the dirac delta is that, for ex, ∫f(x)*delta(x-a) dx from x = -infinity to infinity = f(a). In other words, we solve the equation x-a = 0 for x. Likewise, we have ((x/y)-t) = 0, solved for x = ty.

So I can see where ty*θ(ty) comes from...but how is the |y| derived?

Thank you very much.

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# Homework Help: Integral of x*delta((x/y)-t) dx from 0 to infinity

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