If I have a function $f(t)=\sinh(t)\cos(t)$ how would I go about finding the Laplace transform? I tried putting it into the integral defining Laplace transformation:
$$
F(s)= \int_0^\infty \mathrm{e}^{-st}\sinh(t)\cos(t)\,\mathrm{d}t
$$
But this integral looks very hairy to me. Can $\sinh(t)\cos(t)$ be rewritten as something more manageable perhaps?
[Math] Laplace transform of product of $\sinh(t)$ and $\cos(t)$
hyperbolic-functionslaplace transform
Best Answer
HINT: $$\sinh t\cos t=\frac{e^t-e^{-t}}2\cdot\frac{e^{it}+e^{-it}}2=\frac{e^{t(1+i)}+e^{t(1-i)}-e^{t(i-1)}-e^{-t(1+i)}}4$$
Now, $$L\{e^{at}\}=\frac1{s-a}$$