For the cosine part, we have:
$$
\mathcal{L}\{t \cos(4t)\} = \int_0^\infty t \cos(4t) e^{-st} dt
$$
For the sine part, we have:
$$
\mathcal{L}\{t \sin(3t)\} = \int_0^\infty t \sin(3t) e^{-st} dt
$$
Now, we can use integration by parts to solve these integrals.
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