Answer
$=\frac{1}{2s}-\frac{s}{2(s^2+4a^2)}$
Work Step by Step
Given: $f(t)=\sin^2 at$
Obtain: $F(s)=\int^{\infty}_0 e^{-st}f(t)dt\\
=\int^{\infty}_0 e^{-st} (\sin^2 at)dt\\
=\frac{1}{2}[\lim \int^{\infty}_0 e^{-st}dt-\int^n_0 e^{-st}\cos 2at dt]\\
=\frac{1}{2s}-\frac{1}{2}\lim \int^{\infty}_0 e^{-st}\cos 2at dt\\
=\frac{1}{2s}-\frac{s}{2(s^2+4a^2)}$
