Negative frequency in the Fourier Transform
I think you may be confused about the use of negative frequencies and the sign of the integration coefficient.

Let's take a look at the inverse Fourier Transform

$$ x(t) = \int_{-\infty}^{+\infty}F(\omega)\cdot e^{ 2 \pi i  \omega} d \omega $$

That basically means that you can construct any time domain signal as the from a set of complex exponential. So you can break down a complicated signal into a number of constituent parts that are hopefully easier to work with.

That's typically called an orthogonal base. Complex exponentials are not the only choice, you could use a orthogonal polynomials, wavelets, etc. Complex exponentials are popular since they are easy to work with, specifically when you are dealing with linear time invariant systems

Note two things


The integration goes from $-\infty$ to $+\infty$, i.e you need both negative and positive frequencies to generate any possible time domain signal
The integral variable is positive, but that does only mean than you first integrate the negative frequencies and then the positive ones.


Of course you need to know what the weights for the complex exponentials, so you need to invert the formula above. You get

$$F(\omega) = \int_{-\infty}^{+\infty}x(t)\cdot e^{ -2 \pi i  \omega} d t$$

That's just how the math works out. The integration variable is negative, but that doesn't mean anything. The formula needs to produce results for both positive and negative values of \omega since we need both positive and negative frequencies to represent a time domain signal.