Integrals of products of sines and cosines with arbitrary periods

Question

I am currently studying the Fourier series, which involves integrals of products of sine and cosine functions. Because sine and cosine are orthogonal, we have been using the following facts to help us find the Fourier series of various functions.

$\int_{-\pi}^{\pi}\cos(mx)\cos(nx)dx = \cases{\pi & , $m = n$ \\ 0 & , $m \neq n$} \\ \int_{-\pi}^{\pi}\sin(mx)\sin(nx)dx = \cases{\pi & , $m = n$ \\ 0 & , $m \neq n$} \\ \int_{-\pi}^{\pi}\cos(mx)\sin(nx)dx = 0, \text{for all } m, n$

It seems like there's a more general version of these facts, for sines and cosines with periods other than $2\pi$; e.g. the function $\sin(mx)$ with period $2\pi/m$.

But, I can't seem to find these more general versions anywhere. Could someone please state them here?

Here is an example that seems to come from the more general version of these facts:

$\int_{0}^{3}\sin(\pi x)\sin(\pi x)dx = 3/2$

Answer 1

I think this is what I was looking for:

$\int_{0}^{L} \sin(\frac{\pi x}{L} m) \sin(\frac{\pi x}{L} n) dx = \cases{L/2 & , m = n \\ 0 & , $m \neq n$}$

which came from this video.

Answer 2

I am interpreting your question as you are trying to find which $n$ satisfy $$\int_0^3\sin(\pi x)\sin(n\pi x/3)dx=3/2$$ to do so, we preform the sub. $\pi x/3=u\Rightarrow dx=\frac3\pi du$: $$\frac3\pi\int_0^\pi\sin(3u)\sin(nu)du=\frac32$$ So we are now trying to solve $$\int_0^\pi\sin(3x)\sin(nx)dx=\frac\pi2$$ since $\sin(3x)\sin(nx)$ is symmetric about the $y$-axis, we have that $$\int_0^\pi\sin(3x)\sin(nx)dx=\frac12\int_{-\pi}^{\pi}\sin(3x)\sin(nx)dx$$ so we have to solve $$\int_{-\pi}^{\pi}\sin(3x)\sin(nx)dx=\pi$$ So from the second formula you listed, $$n=3$$ If there's anything I can do to improve my answer, please tell me.