LTI system, Fourier transform, unit step function, sine wave
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 $\cos(\omega_0t)$