00:01
Hello, in this case we need to show that this integral here is equal to zero.
00:06
Okay, so to do this we need to use some trigonometric identities.
00:15
So in this case, i'm going to regret this as the integral between minus pi and pi one half two times the cosine of m x, the cosine of n x, the cosine of n x, the x okay so this part here is equal i hope i have enough space so it's one half of the integral minus pi pi of the cosine of m plus n plus the cosine of the cosine of the cosine of the cosine of m minus n times x d x okay so we got this integral here one half and well what i have done with this using this trigonometric identity is that i separate this multiplication that could complicate the things because in that case we need to use integration by parts so maybe a couple of times until we obtain some kind of formula to solve this integral.
01:51
However, if we transform this multiplication into the summation of two cosines, well this is easy to integrate.
01:59
So the integral in this case is the sign of m plus n x divided m plus n minus pi pi plus the the sign of m minus n x m minus n between minus pi and pi okay so here we got something m and n are positive integers okay and we know that the sign of any integer alpha times pi will be equal to zero okay this is based on some geometric intuition the sign of at this point and at this point let's put some axis here x and y so if we choose this point here that corresponds to zero or pi or two pi or three pi or four pi no matter what integer times pi the design will be always located at this point of on the circle and that corresponds to zero so the sign of any integer alpha times pi will be zero and what we have here is that because m plus n is also an a positive integer and m minus n will be an integer.
03:53
It could be negative but it doesn't matter.
03:56
For any integer here multiplying pi to the sign that will be zero.
04:01
So these these evaluations here after integrating are equals to zero.
04:13
Okay so these are equals this is equal to zero and this is also equals to zero.
04:19
So the solution for this integral is that the integral of minus pi pi cosine m x cosine n x the x is equal to zero and here we have shown okay so that is the first part now we need to show the same but with instead of using cosines using signs so the second part of the exercise is minus pi pi sine here is sign of mx, sine nx, d x .x, great.
05:16
So we're going to use again an identity that is really similar to the previous one...