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in this video were going to look at a pretty interesting infinite integral and that will be the integral from 0 to infinity of the absolute value of sine of x times sine of X over X DX so I think its a well-known fact that if we were to remove this absolute value of sine of X that integral would be PI over 2 and well actually get to prove that as well because were going to approach this integral using something called lobha jetskis integral formula which allows us to evaluate a whole family of integrals that are of this form and in order to prove that formula were going to need some ideas from Fourier analysis and lets look at this one first so if f of X has a period of 2 pi then we can write f of X as this doubling infinite sum and equal negative infinity to infinity of CN times e to the I in X where CN is 1 over 2 pi times the integral from minus PI to PI of f of X e to the minus I and X DX so thats how you calculate those coefficients so this is a fact that were not going t