Fourier transform, how should I solve there one?

Question

I can't figure out how to solve this problem, could anyone help me?

Find the Fourier transform of F(x) = { sin(x), if |x| <= pi and 0(null) if, |x| > pi }

*edit: I got stuck on the following integral

Please, explain if it's possible Thanks in advance!

Answer 1

Hint: The Fourier transform is defined as

$$ F(s) = \int_{-\infty}^\infty e^{-ist} f(t) \ dt $$

As $f$ is piecewise defined, you have really one integral to deal with:

$$ F(s) = \int_{-\pi}^{\pi} e^{-ist} \sin t \ dt $$

Now you have two options: write $e^{-ist}$ in terms of sines and cosines, or use a nice even-odd trick. Can you take it from here?

Answer 2

This is a rectangle function multiplied by a sine wave. So, to get the ft, you can convolve the fourier transform of the rectangle (a sinc function) with the fourier transform of the sine wave (a sum of 2 delta functions) and use the delta function's sifting property to gettheresult.