# Thread: Fouries series integration help

1. ## Fouries series integration help

Hey everyone, I am stuck on 2 questions, so if anyone can help me that would be much appreciated.

$\displaystyle f(x)=\begin{Bmatrix} 2, & -1<x<1\\0, & 1<|x|<2\end{matrix}$

Fourier Series representation:
$\displaystyle f(x)=a_0 + \sum_{n=1}^{\infty}(a_n \cos(\frac{n \pi x}{2}) +b_n \sin(\frac{n \pi x}{2}))$

The first question is to integrate both sides of $\displaystyle f(x)$ from -2 to 2 to show that $\displaystyle a_0=1$.
The second question is to multiply both sides of $\displaystyle f(x)$ by $\displaystyle sin(\frac{m \pi x}{2})$ and integrate from -2 to 2 to show that $\displaystyle b_n=0$.

Because of the domain restrictions on f(x) I can't figure out what to do for these questions.

If you can help me it would be nice. Thanks

2. Integrating F equation
$\displaystyle \int_{-2}^2 f(x) \; dx=\int_{-2}^2 a_0 \; dx$
May be shown that other terms are zeros.
$\displaystyle \int_{-2}^2 f(x) \; dx=\int_{-1}^1 2 \; dx=4$
$\displaystyle \int_{-2}^2 a_0 \; dx=4a_0$

3. Thanks zzzoak, I get question 1, but I still don't understand how to do the 2nd question.