00:01
Hello everyone.
00:02
We are asked to find the fourier series of the given function f of x is equal to 3x in the interval minus pi less not equal to x less than 0 and 0 in the interval 0 less than pi.
00:17
And the function as a period 2 pi because f of x comma x plus 2 pi is equal to f of x implies that it is of period 2 pi which is equal to f of x implies that it is of period 2 pi which is equal to 5.
00:34
To 2l.
00:35
So on comparison we see that l is nothing but.
00:41
So therefore, we find the four year coefficients that is a0, a .n and bn.
00:48
So let's proceed to the solution.
00:52
So first of all, the formula for a0 is given by 1, divot by l integral over 2l, f of x, dx.
01:05
Now since l is nothing.
01:07
But pi, we can write this to be a not is equal to 1 divided by pi integral minus pi to 0 tx dx plus integral 0 to pi 0 d x as given the question it is piecewise so we are writing it like this so so we have integral 0 to pi 0 d x can always be neglected because it is always going to be equal to 0.
01:41
Therefore, i just need to calculate the value of this for a and bn also.
01:47
So therefore, we have a0 to be equal to 1 divided by pi.
01:54
Upon integration, i'm going to get 3x squared by 2 minus pi to 0.
02:02
So let's substitute the limits.
02:06
We will get 0 minus 3 minus pi the whole square divided by 2.
02:16
So this is going to leave me the answer.
02:18
A0 is equal to 3 pi by.
02:27
So now let's proceed to find a .n.
02:31
So a .n is given by a .n.
02:36
Is equal to 1 divided by l minus l to l, f of x, pi x divided by l dx.
02:50
We know that l is starting but pi.
02:54
So let's substitute the value of l and the value of f and find the a .n.
03:00
So a .n will be equal to 1 divided by pi integral minus pi to pi to pi.
03:07
But we know that minus pi to 0 only we have 3x.
03:12
So we'll put minus pi to 0 and 0 to pi is nothing but again 0.
03:17
So we neglect that.
03:20
N, pi x...