laplace transformation of cosat divided dy t

Question

laplace transformation using property division by t

Answer 1

f(t)=cos(at)/t, f'(t)=-asin(at)/t-cos(at)/t².

ℒ{f(t)}=∫[0^-,∞]e^-stf(t)dt=∫[0^-,∞]e^-st(cos(at)/t)dt.

Let u=f(t), du=f'(t)dt, dv=e^-stdt, v=-e^-st/s, then integrating by parts:

ℒ{f(t)}=-e^-stf(t)/s+(1/s)∫e^-stf'(t)dt=f(0^-)/s+(1/s)ℒ{f'(t)} (integral evaluated for t∈[0^-,∞]).

f(0^-)=1/t→∞; similarly f'(0^-)→∞. This precludes evaluation of the transform.