00:01
Hello student.
00:01
So in this problem we are going to evaluate the fourier transform of equation x of t is equal to t square.
00:10
So in order to take a fourier transform we should write the formula for fourier transform that is capital x of x is equal to minus infinity to plus infinity small x of t into exponential of i s t dt.
00:35
So this is the fourier transform of a given equation.
00:39
So in order to do that we should substitute t square instead of x of t.
00:46
So the equation becomes integration of minus infinity to plus infinity t square e power i s t dt.
00:59
So now we are going to separate the limits into three parts that is which is equal to exponent integration of minus infinity to minus a t square exponential of i s t dt plus integration of minus a to plus a t square exponential of i s t dt plus integration of plus a to plus infinity t square e power i s 2 t.
01:50
So here this part should be is equal to zero.
01:55
Similarly this integration also has the value of zero.
02:01
So we have only one part that is minus a 2 plus a t square e power i s t dt.
02:14
So now we are going to evaluate this equation with the help of bernoulli's rule.
02:22
So we can write the bernoulli's equation that is integration of u v dx is equal to u integration of v dx minus u prime double integral of v dx dx plus u double prime triple integral v dx dx dx.
02:56
So using this bernoulli's rule so here t square is u and this is e power i x t is so the equation becomes that is x of x is equal to t square e power i s t divided by i s minus 2 t e power i s t divided by i s the whole square plus 2 e power i s t divided by i s the whole.
03:52
So here the limit is minus a 2 plus a.
04:00
So here we have i s the whole power square.
04:03
So i square is equal to one.
04:05
So by simplifying this equation we get t square e power i s t divided by i s.
04:16
So here we have i square right.
04:19
So this i square is equal to minus one...