# Laplace Transform of t cos(2t)

## Homework Statement

Find the Laplace Transform from the definition of f(t) = tcos(2t)

## Homework Equations

$$\int e^-^s^ttcos(2t)dt$$

## The Attempt at a Solution

I started by doing parts
$$u = t$$
$$du = dt$$

$$dv = cos(2t)e^-^s^t dt$$

but I get stuck on v and as far as I can tell doing parts on v won't help because neither term will be reduced when differentiated.

I'm looking for some direction on how to solve this integral.
Thanks!

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vela
Staff Emeritus
Homework Helper
You might try splitting it up differently. Try

\begin{align*} u & = t e^{-st} \\ dv & = \cos 2t \end{align*}

You'll have to integrate by parts at least twice.

Thanks. I was able to figure it out after starting with it split how you suggested.

A sneaky way is to use the fact that $L[t f(t) ] = - F^{\prime}(s)$. You can use the fact that $L[ \cos (2t) ] = \frac{s}{s^2 + 4}$ for s>0. This way you are taking a derivative instead of integrating!

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