I am trying to calculate one FT problem, but when I finished this in my way, and then looked the answer, I found I probably derive an incorrect answer. However, I can not figure out which steps are wrong. Any suggestions are appreciated
$$x(t) = e^{j3t}e^{-|5t+7|}$$
My method:
1.Rewrite $$x(t)$$ as $$x(t) = e^{-7}e^{j3t}e^{-5t}u(t) + e^{7}e^{5t}u(-t)e^{j3t}$$ where u(t) is unit step function
2.Define helper functions $$v_1(t) = e^{-5t}u(t) \to V_1(w) = \frac{1}{5+jw}$$ $$v_2(t) = v_1(-t) \to V_2(w) = \frac{1}{5-jw}$$ $$v_3(t) = e^{j3t}v_1(t) \to V_3(w) = \frac{1}{5+j(w-3)}$$ $$v_4(t) = e^{j3t}v_2(t) \to V_4(w) = \frac{1}{5-j(w-3)}$$
3.Combine above functions together $$X(w) = \frac{e^{-7}}{5+j(w-3)} + \frac{e^{7}}{5-j(w-3)}$$
The correct answer is: $$X(w) = \frac{10e^{\frac{j7(w-3)}{5}}}{(w-3)^{2}+25}$$
Your error is right at the start: your rewritten expression for $x(t)$ is wrong.
We have $5t+7$ changes sign when $t=-7/5$, not when $t=0$. So your step functions need to be shifted.