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## Homework Statement

Compute the average value of the function:

f(x) = δ(x-1)*16x

^{2}sin(πx/2)*e

^{iπx}/(1+x)(2-x)

over the interval x ∈ [0, 8]. Note that δ(x) is the Dirac δ-function, and exp(iπ) = −1.

## Homework Equations

∫ 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 16x

^{2}sin(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.

Many thanks.