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

Evaluate the following integrals:

[tex]\int^{+\infty}_{-\infty}\delta[f(x)]dx[/tex]

and

[tex]\int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx[/tex]

2. Relevant equations

[tex]\int^{+\infty}_{-\infty}\delta(x)dx=1[/tex]

[tex]\int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0)[/tex]

[tex]\int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)[/tex]

3. The attempt at a solution

Part a:

[tex]\int^{+\infty}_{-\infty}\delta[f(x)]dx[/tex]

let:

[tex]u=f(x)[/tex]

[tex]du=f'(x)dx[/tex]

[tex]\int^{+\infty}_{-\infty}\delta[f(x)]dx=\int^{+\infty}_{-\infty}\frac{\delta(u)du}{f'(x)}=\int^{+\infty}_{-\infty}\frac{\delta(u)du}{f'[f^{-1}(u)]}=\frac{1}{f'[f^{-1}(0)]}[/tex]

Is this correct?

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# Homework Help: Dirac Delta Function

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