My result contains 'limit' value

My code is

clc;

syms x t X omega

a = 2; b = 2;

x = exp(-a*t)*sin(b*pi*t);

X = simplify(int(x*exp(-j*omega*t),t,0,inf))

in books, its result is

X =

2*pi/(4-w^2+4*pi^2+4*i*w)

however I got

X =

(2*pi)/((2 + omega*1i)^2 + 4*pi^2) - limit(exp(- 2*t - t*omega*1i)*(sin(2*t*pi)*(2 + omega*1i) + 2*pi*cos(2*t*pi)), t, Inf)/((2 + omega*1i)^2 + 4*pi^2)

how can I get the answer on the book?