Recall from the Double Angle Formula that

`sin 2α = 2\ sin α\ cos α`

We can use this to re-express our integrand (the part we are integrating):

`sin at\ cos at=1/2 sin 2at`

So the Laplace Transform of the integral becomes:

`Lap{int_0^t\ sin at\ cos at\ dt}=1/2 Lap{int_0^t\ sin 2at\ dt}`

`=1/2(2a)/(s(s^2+4a^2))`

`=a/(s(s^2+4a^2))`

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