Compute the average value of the function:
f(x) = δ(x-1)*16x2sin(πx/2)*eiπx/(1+x)(2-x)
over the interval x ∈ [0, 8]. Note that δ(x) is the Dirac δ-function, and exp(iπ) = −1.
∫ dx δ(x-y) f(x) = f(y)
The Attempt at a Solution
Average of f(x) = 1/8 ∫from 0 to 8 δ(x-1) dx 16x2sin(πx/2)*(-1)/(1+x)(2-x)
Average of f(x) = -1
Is this correct? I'm unsure of whether you can just use δ(x-a) = δ(x-1) and let a=1 and not let a=-1? I don't get how to use this bit of the function as I seem to have just ignored the negative sign.