Answer to Question #152814 in Differential Equations for Kumar

Question

 Find the Fourier Sine transform of f(x) = 1 in   0,l

Answer 1

$\displaystyle \mathcal{F}(f(x)) = \sqrt{\frac{2}{\pi}}\int_0^\infty f(x)\sin(\omega x) \, \mathrm{d}x\\ \begin{aligned} \mathcal{F}(1) &= \sqrt{\frac{2}{\pi}}\int_0^l 1 \times \sin(\omega x) \, \mathrm{d}x = \sqrt{\frac{2}{\pi}}\int_0^l \sin(\omega x) \, \mathrm{d}x \\&= \sqrt{\frac{2}{\pi}}\int_0^l \frac{-\cos(\omega x)}{\omega} \, \mathrm{d}x \\&= \sqrt{\frac{2}{\pi}}\cdot \frac{-\cos(\omega x)}{\omega}\biggr\vert_0^l = \sqrt{\frac{2}{\pi}} \cdot \frac{\cos(\omega x)}{\omega}\biggr\vert_l^0 \\&= \sqrt{\frac{2}{\pi}}\left(\frac{1 - \cos(l\omega)}{\omega}\right) \end{aligned}$