MHB Help with Fourier series mistake

[ognik]

Hi - frustratingly I get some problems right 1st time, others just defy me (Headbang)

$f(x) = -x, [-\pi,0]; = x, [0,\pi]$

I get $a_0 = \pi$ and $a_n = \frac{-4}{\pi \left(2n-1\right)^2}$ which agrees with the book - but I thought I'd check $b_n$ for practice, it should = 0 according to the book, but I got:
$ \frac{1}{\pi} \left[ \int_{-\pi}^{0}-x Sinnx \,dx + \int_{0}^{\pi}x Sin nx \,dx \right] $
$= \frac{2}{\pi}\int_{0}^{\pi}x Sin nx \,dx $
$ = \frac{2}{\pi}\left[ \frac{x}{n}\left(-Cosnx\right) \right]^{\pi}_0 + 0 \ne 0$?

 

[Dethrone]

I believe your second step is faulty - you can't just double the integral because they can't be combined. I think you need both parts so they somehow cancel each other.

 

[ognik]

...and so they do, thanks