Euler equation formula

Question

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$

I am getting following integrals:

$\int_{-\infty}^{\infty} dx f(x) cos(kx)$ (for the real part)

and

$i* \int_{-\infty}^{\infty} dx f(x) sin(kx)$ (for its imaginary part)

I am wondering what is the final integration result though. Is that the sum of both parts or are they seperate results?

Answer 1

Integration is linear (that is integral of sum is sum of integrals) so yes, final result is sum. I think most od the time it's not convenient to split integrals this way though, because exponents are easier to handle than trig functions.