Half range sine series is given by

$b_{n} = \frac{2}{l}\int\limits_{0}^{l}f(x)sin\bigg(\frac{n\pi x}{l}\bigg) dx$ a = 0 & $a_{n}$ = 0

Here $f(x) = x^2$ and l = 3

$b_{n} = \frac{2}{3}\int\limits_{0}^{3}x^2sin\bigg(\frac{n\pi x}{3}\bigg) dx$

Using Leibnitz rule,

$b_{n} = \frac{2}{3}\bigg[x^2 \frac{\bigg(\frac{-cosn\pi x}{3}\bigg)}{\frac{n\pi}{3}} - \frac{(2x)\frac{-sinn\pi x}{3}}{\bigg(\frac{n\pi}{3}\bigg)^2} + \frac{2\bigg(\frac{cosn\pi x}{3}\bigg)}{\bigg(\frac{n\pi}{3}\bigg)^3}\bigg]_{0}^{3}$

$b_{n} = \frac{2}{3}\bigg[3^2 \frac{3}{n\pi}(-cosn\pi) …