Question

Find the transform and sketch the correct graph clearly I will upvote

Select the correct graph of the given function, which is assumed to be zero outside the given interval. Then find its transform. cos (8t) (0 < t <a) у 0.51 -0.51 t -14 TT 31 TT 4 2 4

Answer 1

By transform if you mean Laplace transform then

Given function is $f(t)=\cos(8t)~~~~,~~~0<t<\pi$

By definition of Laplace transform we get

$\frak{L} (f(t))=\int_{0}^{ \infty} f(t) e^{-st} dt \\ \\~~~ ~~~~~~~~~~~~~= \int_{0}^{\pi}\cos (8t)e^{-st }dt\\~~~ ~~~~~~~~~~~~~= \left [ {e^{-st}\over 8^2+(-s)^2}[8\sin(8t)+(-s)\cos (8t)] \right ]_{0}^{\pi} \\ \\~~~ ~~~~~~~~~~~~~={s[1-e^{-\pi s}]\over s^2+64}$