Im trying to find the Fourier Transform of $ f(x) = \sin(x) \cdot e^{-|x|}$
I applied the standard formula and got to this point:
$\tilde{f(t)}= \frac{1}{\sqrt{2\pi}}\cdot \int_{-\infty}^{+\infty}\sin(x)e^{-|x|}e^{-itx} dx $
= $\frac{1}{\sqrt{2\pi}}\cdot \int_{-\infty}^{0}\sin(x)e^{x}e^{-itx} dx+\frac{1}{\sqrt{2\pi}}\cdot \int_{0}^{+\infty}\sin(x)e^{-x}e^{-itx} dx $
How can I integrate this - I tried integration by parts and did not get anywhere because there will always be alternation between sin and cos and also there is an e function in the integrand.
Now Im trying to think about integrating by the help of substitution but I couldn't think of anything to substitute it with. Can you please help :) ?
Thanks a lot!