Find Fourier Series of the function $f(x)=\sin x + \cos x$

Question

I calculated the fourier series of the given function, but all of the fourier coefficient was 0, it sounds strange!

Answer 1

$f(x)$ is already in Fourier Series representation.

But if you want to do the integral to get the coefficients:

$$ a_1=\frac{1}{\pi}\int_0^{2\pi}\left(\left(\sin x+\cos x\right)\cdot\cos\left(x\right)\right)dx = 1\\ b_1=\frac{1}{\pi}\int_0^{2\pi}\left(\left(\sin x+\cos x\right)\cdot\sin\left(x\right)\right)dx = 1 $$