Integrating exponent to get delta function

[tamiry]

Something i ran into while doing hw

Homework Statement

starting with
$$\int{dx} e^{-ikx}\delta(x) = 1$$
we conclude by Fourier theory that
$$\int{dk} e^{+ikx} = \delta(x)$$
Now, i try to compute
$$\int{dk} e^{-ikx}$$

(I've dropped the normalization factors of $$2\pi$$. I believe no harm is done by that)

Homework Equations

The Attempt at a Solution

Method 1: change x to -x
$$\int{dk} e^{-ikx} = \int{dk} e^{+ik(-x)} = \delta(-x) = \delta(x)$$

Method 2: change the integration parameter k to -k
$$\int{dk} e^{-ikx} = -\int{dk} e^{+ikx} = -\delta(x)$$

So what did I do wrong here?


thanks a lot
T

 

[tamiry]

I looked at it again. At my second method. I had to change the integral limits as well, and that fixes it.