Answer to Question #189381 in Differential Equations for Aniket

Question

Fourier cosine integral of f(x) is

A)

cos Ax ff(t)sina tdt da 0

d cos 2x ff(t) cos at dt da

B) cosax ff(t)sintdtdx

D) 25 fcos Ax f f(t)cos At dt da

Answer 1

$\begin{aligned} \mathcal{F}(f(x)) &= \sqrt{\frac{2}{\pi}}\int_0^a \cos(at)\cos(\omega t) \,\,\mathrm{d}t\\ &= \frac{1}{\sqrt{2\pi}}\int_0^a \cos((a - \omega)t) + \cos((a + \omega)t) \mathrm{d}t \\&= \frac{1}{\sqrt{2\pi}}\int_0^a \frac{\sin((a - \omega)t)}{a - \omega} + \frac{\sin((a + \omega)t)}{a + \omega}\mathrm{d}t \\&= \frac{1}{\sqrt{2\pi}}\frac{\sin((a - \omega)a)}{a - \omega} + \frac{\sin((a + \omega)a)}{a + \omega}\mathrm{d}t \end{aligned}$