Laplace Transform of: $\cos(4t) * \sin(3t)$

Question

I am new to Laplace Transform so Can anyone please solve this question ? $$ \mathcal{L}\left[ \cos( 4t ) * \sin( 3t )\right] $$

Answer 1

HINT

One of the famous properties of the Laplace transform is that it converts convolution into multiplication, so one step may be $$ \mathcal{L}\left[ \cos( 4t ) * \sin( 3t )\right] = \mathcal{L}\left[ \cos( 4t )\right] \cdot \mathcal{L}\left[ \sin( 3t )\right] $$

Can you take it from here?

Answer 2

In Mathematica you would write

Clear[t, s, m];
Integrate[Cos[4*t]*Sin[3*t]*Exp[-s*t], {t, 0, Infinity}]

which gives the output:

ConditionalExpression[(3 (-7 + s^2))/(49 + 50 s^2 + s^4), Re[s] > 0]

which is the same as:

LaplaceTransform[Cos[4*t]*Sin[3*t], t, s]

which gives the output:

(3 (-7 + s^2))/(49 + 50 s^2 + s^4)