obtain the fourier series expansion of f(x) = 1+x if -2<x<0 ; f(x) = 1-x if 0<x<2

Answer 1

Last time I did anything like was over 60 years ago.
Found this video that almost parallels your problem, (I had difficulty understanding
him, perhaps that is why they have subtitles)

https://www.google.ca/search?rlz=1C6CHFA_enCA690CA691&sxsrf=ALeKk00pcGpMsTZFud1k9gLG2mAxwA07Ag%3A1584971926465&source=hp&ei=lsB4XoLzGPK1gge5qKX4BA&q=fourier+series+expansion&oq=fourier+series+expansion&gs_l=psy-ab.3..0l10.2404.2404..2786...2.0..0.78.78.1......0....2j1..gws-wiz.....10..35i362i39.8jmHAy3SNl0&ved=0ahUKEwiClpzF4LDoAhXymuAKHTlUCU8Q4dUDCAY&uact=5#kpvalbx=_msB4XpqKJcHctQbCtou4DA68

Answer 2

You can check your work at any of a number of online FS calculators, such as the one at

www.symbolab.com/solver/fourier-series-calculator