Questions about the delta function

[rmiller70015]

Homework Statement

I just have a quick question about the delta function, I'm pretty confident in most other cases but in this simple one I'm not so sure.

$$\int_{-\infty}^{\infty} \phi (x)\delta (-x)dx$$

Homework Equations

The Attempt at a Solution

[/B]
$$\int_{-\infty}^{\infty} \phi (x)\delta (-x)dx$$
Using substitution where u=-x and du=-dx:
$$\int_{-\infty}^{\infty} \phi (-u)\delta (u)(-du) = -\phi (0)$$
Is this correct?

 

[BvU]

Wouldn't that yield $\phi(-0)$ instead of $-\phi(0)$ ?

 

[rmiller70015]

Wouldn't that yield $\phi(-0)$ instead of $-\phi(0)$ ?

I'm not entirely sure. I do realize that there should be a negative inside the phi test function but I omitted it because it's zero anyway and I would have had to account for it if the delta function was something like (x-a), but here I didn't. However, when I do the substitution I have to deal with a negative u differential which makes the whole function negative in my mind.

 

[BvU]

And the bounds of the integral ?

 

[rmiller70015]

Ahh the bounds so when I do the substitution I get:
$$-\int_{\infty}^{-\infty} \phi (-u)\delta (u)du$$
Then I change my limits and lose the negative.
Thank you.