Question

The Fourier series expansion is given as: 
\[ \frac{a_{0}}{2} + a_{1}\cos x + a_{2}\cos 2x + \cdots + b_{1}\sin x + b_{2}\sin 2x + \cdots \] 

For the function \[ f(x) = x + \frac{x^{2}}{4}, \quad -\pi \le x \le \pi \] 
The value of \(a_0\) and \(b_2\) are:

• A. $a_{0}=\frac{3\pi^{2}}{5},b_{2}=-2$
• B. $a_{o}=\frac{\pi^{2}}{6},b_{2}=+1$
• C. $a_{o}=\frac{\pi^{2}}{6},b_{2}=-1$
• D. $a_{0}=\frac{5\pi^{2}}{3},b_{2}=+2$

Hint:

When the interval is symmetric $[-\pi, \pi]$, always check if the function is Odd or Even. Integral of an Odd function over this interval is always 0, saving you significant calculation time.

Answer 1

The Correct Option is C