I want to find the nth order Fourier approximation for $\displaystyle f(x)=x$. Since this function is odd, the projections on all cosines will be zero, hence it will be expressed through the sines only. So I just need to find the sine coefficients.

The problem is that I checked the answer to this problem in two different textbooks, the solutions in the books are totally different from each other.

$\displaystyle b_k= \frac{1}{\pi} \int^{2 \pi}_{0} f(x)sin(kx)= \frac{1}{\pi} \int^{2 \pi}_0 xsin(kx) dx =- \frac{2}{k}$

- The first book calculates the coefficients like this:

$\displaystyle b_k = (f(x),sin (k \pi x))= \int^1_{-1} x sin (k \pi x) dx= -x\frac{cos (k \pi x)}{k \pi} ]^1_{-1} + \int^1_{-1} \frac{cos (k \pi x)}{k \pi}dx $$\displaystyle = -2 \frac{cos k \pi}{k \pi}=-(-1)^k \frac{2}{k \pi}$

- The second book calculates the coefficients this way:

So which one is correct? I mean, they both are probably correct but I don't understand why they use different methods and end up with different answers. Any explanation is very appreciated.