# Thread: Fourier Series Integration Fourmula

1. ## Fourier Series Integration Fourmula

This is one of the example from the book named Higher Engg. Mathematics III by Dr. Kachot
Obtain the fourier series for:
$\displaystyle f(x)=\frac{1}{4} (\pi-x)^2 ; 0\leq x \leq 2\pi \\=f(x+2\pi);otherwise$

Here the value of
$\displaystyle a_{0}=\frac{1}{\pi}\int_{0}^{2\pi} f(x)dx = \frac{1}{\pi}\int_{0}^{2\pi} \frac{(\pi-x)^2}{4}dx = \frac{1}{4\pi}[-\frac{(\pi-x)^3}{3}]_{0}^{2\pi}$

Please explain the above step. Didn't get which integration formula the author applied. Also the minus sign is bugging me. Where did it come from?

2. ## Re: Fourier Series Integration Fourmula

Minus Sign - Think Chain Rule.

$\displaystyle \int x^2\;dx = \frac{x^{3}}{3} + C$

3. ## Re: Fourier Series Integration Fourmula

Chain rule? Didn't get you

4. ## Re: Fourier Series Integration Fourmula

Originally Posted by maulin
Chain rule? Didn't get you
Make the substitution

$\displaystyle u=\pi-x$

$\displaystyle \frac{du}{dx}=-1\Rightarrow\ dx=-du$

Then the integral becomes

$\displaystyle -\frac{1}{4{\pi}}\int_{x=0}^{x=2{\pi}}u^2du$

5. ## Re: Fourier Series Integration Fourmula

Originally Posted by maulin
Chain rule? Didn't get you
$\displaystyle \frac{d}{dx}f(x) = f'(x)$

$\displaystyle \frac{d}{dx}f(-x) = -f'(-x)$

6. ## Re: Fourier Series Integration Fourmula

Thanks, now I got it!