Laplace transform for $-t\cos(2t)$

Question

This Laplace transform exercise is giving me a headache. I was trying to use the definition of the Laplace transform but when I make the $u$ and $dv$ substitutions for the integration by parts I never find the original integral, like in the other exercises, and just go on an endless loop. What am I doing wrong?

Answer 1

Hint: the Laplace transform of a function $-tf(t)$ is the derivative of the Laplace transform of $f$ (say for functions with exponential growth, such as yours).

Answer 2

The easiest approach is to write $\cos(2t) = \frac{e^{2it}+e^{-2it}}{2}$ and compute $\int_0^\infty te^{-st\pm2it}dt$ using integration by parts.

Answer 3

One may observe that $$ \int_0^\infty \cos(2t)e^{-st}dt =\Re\int_0^\infty e^{-(s+2i)t}dt=\Re\frac1{s+2i} =\frac{s}{s^2+4}, $$ then differentiating with respect to $s$, we get

$$ -\int_0^\infty t\cos(2t)e^{-st}dt =\frac{4-s^2}{\left(s^2+4\right)^2}. $$