LTI system, Fourier transform, unit step function, sine wave I just finished learning about Fourier Transforms and don't understand this signal: 

$$x(t) = \cos(\omega t)u(t) $$

This is a cosine wave but only where $\omega$ is positive. My question is what can I do to figure out the output when the impulse response $h(t) = e^{-2\pi t}u(t)$ ? 

I first try to break $x(t)$ up using Euler to get:

$$x(t) = \frac{1}{2} \left[e^{j2\pi t} + e^{-j2\pi t}\right]$$ 

then get FT of each component and use convolution property to get:

$$Y(j\omega) = H(j\omega)X(j\omega)$$

but kept getting an incorrect answer.
I can see three problems here:

1) $x(t)$ is a cosine wave but only where $t$ is positive, not $\omega$

2) You must keep the step function when applying Euler's identity, and I can't see the cosine frequency in your second expression for $x(t)$ either. Always make sure your identity is right, with WolframAlpha for example

3) Do not mix up your $\omega$ that is a parameter to $\cos(\omega t)$ and the $\omega$ variable of the frequency domain in $X(j\omega)$. Use different notations, like $\cos(\omega_0t)$