Question

Find the laplace of cosat where a is a constant

Answer 1

By definition of a Laplace transform we have

$\mathcal{L}(\cos at)(s)=f(s)=\int_0^{+\infty} e^{-st}\cos(at)\: dt$

We will apply the integration by parts :

$f(s)= \frac{1}{a}\left(\left[e^{-st}\sin(at)\right]^{+\infty}_0+\int_0^{+\infty}se^{-st}\sin(at)\: dt \right)$

The first bracket is 0, as $e^{-st}\to0 \text{ when }t\to +\infty$ and $\sin(at)_{t=0}=0$. We then apply the integration by parts to the integral :

$f(s)=\left[-\frac{s}{a^2}e^{-st} \cos(at)\right]^{+\infty}_0-\frac{s^2}{a^2} \int_0^{+\infty} e^{-st} \cos(at) \: dt$

The first bracket gives $\frac{s}{a^2}$ and the integral is just $f(s)$, so we have :

$f(s)=\frac{s}{a^2}-\frac{s^2}{a^2}f(s)$

$\frac{a^2+s^2}{a^2}f(s)=\frac{s}{a^2}$

And therefore, the final answer is $f(s)=\frac{s}{s^2+a^2}$ (defined for $s>0$).