$\displaystyle l=2 \\[2ex] \displaystyle For\ Half\ range\ sine\ series,\ b_n=\frac{2}{l}\int_0^lf\left(x\right)\ \sin ⁡(\frac{n\pi{}x}{l}) dx \\[2ex] \displaystyle b_n=\frac{2}{2}\int_0^2x\left(2-x\right)\sin {\left(\frac{n\pi{}x}{2}\right)}\ \ dx \\[2ex] $

$\displaystyle \int x\left(2-x\right)\sin{\left(\frac{n\pi{}x}{2}\right)}dx\ \\[2ex] \displaystyle=x\left(2-x\right)\int\sin{\left(\frac{n\pi{}x}{2}\right)}dx\ -\frac{d}{dx}\left[x\left(2-x\right)\right]\int\int\sin{\left(\frac{n\pi{}x}{2}\right)}dx\ dx\\[3ex] \ \ \ \ \ \ \ \ \ \displaystyle+\frac{d^2}{dx^2}\left[x\left(2-x\right)\int\int\int\sin{\left(\frac{n\pi{}x}{2}\right)}dx\ dx\ dx\right]-0\ \ \\[2ex] \displaystyle =x\left(2-x\right)\left[-\frac{\cos{\left(\frac{n\pi{}x}{2}\right)}}{\left(\frac{n\pi{}}{2}\right)}\right]-\left(2-2x\right)\left[-\frac{\sin\ {\left(\frac{n\pi{}x}{2}\right)}}{{\left(\frac{n\pi{}}{2}\right)}^2}\right]+\left(-2\ \right)\left[\frac{\cos{\left(\frac{n\pi{}x}{2}\right)}}{{\left(\frac{n\pi{}}{2}\right)}^3}\right] \\[3ex] \displaystyle b_n={\left[x\left(2-x\right)\left[-\frac{\cos{\left(\frac{n\pi{}x}{2}\right)}}{\left(\frac{n\pi{}}{2}\right)}\right]-\left(2-2x\right)\left[-\frac{\sin\ {\left(\frac{n\pi{}x}{2}\right)}}{{\left(\frac{n\pi{}}{2}\right)}^2}\right]+\left(-2\ \right)\left[\frac{\cos{\left(\frac{n\pi{}x}{2}\right)}}{{\left(\frac{n\pi{}}{2}\right)}^3}\right]\right]}_0^2 \\[3ex] \displaystyle b_n=0-\left(2-4\right)\left[-\frac{\sin\ {\left(n\pi{}\right)}}{{\left(\frac{n\pi{}}{2}\right)}^2}\right]-2\left[\frac{\cos{\left(n\ \pi{}\right)}}{{\left(\frac{n\pi{}}{2}\right)}^3}\right]\ -0+2*0+\frac{2}{{\left(\frac{n\pi{}}{2}\right)}^3}\ \\[5ex] \displaystyle now\ ,\sin{n\ pi=0}\ ,\cos{n\ \pi{}}={\left(-1\right)}^n \\[2ex] \displaystyle b_n=-\frac{2{\left(-1\right)}^n}{{\left(\frac{n\pi{}}{2}\right)}^3}+\frac{2}{{\left(\frac{n\pi{}}{2}\right)}^3}=\frac{2^4\left[1-{\left(-1\right)}^n\right]}{n^3{\pi{}}^3} \\[2ex] \displaystyle f\left(x\right)=\sum_{n=1}^{\infty{}}b_n\sin{\left(\frac{n\pi{}x}{l}\right)}\ \\[2ex] \displaystyle f\left(x\right)=\ \sum_{n=1}^{\infty{}}\left[\frac{2^4\left[1-{\left(-1\right)}^n\right]}{n^3{\pi{}}^3}\right]\sin{\left(\frac{n\pi{}x}{l}\right)} \\[5ex] When \ `n' \ is \ even, \ 1-{\left(-1\right)}^n=1-1=\ 0\\[2ex] When \ `n'\ is\ odd, \ 1-{\left(-1\right)}^n=1+1=\ 2\\[3ex] \displaystyle f\left(x\right)=\frac{2^5}{{\pi{}}^3}\sum_{n=1}^{\infty{}}\frac{\sin{\left(\frac{n\pi{}x}{l}\right)}}{n^3\ } \\[3ex] \displaystyle \ n=odd=\ 2k-1,\ k=1,2,3,4…..\ \\[2ex] $