Proving an equation involving integrals and limits

Question

I have to show the following equation:

$\large\int_0^\infty \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\frac{s}{s^2+\beta ^2}$ with $s>0$

I've come so far:

$\large\int_0^\infty \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\lim_{a \to \infty}\int_0^a \! e^{-st}\cos(\beta t) \, \mathrm{d}t=\Large\lim_{a \to \infty}\frac{(\beta\sin(\beta a)-s\cos(\beta a))e^{-sa}}{s^2+\beta ^2}$

the last term is zero, right? But it has to be $\large\frac{s}{s^2+\beta ^2}$

Where is my mistake?

Answer 1

As written your limit it zero, but you're missing a term in the last step, namely

$$-\frac{\beta\sin(\beta0) - s\cos(\beta0)}{s^2 + \beta^2}e^{-s0} =\frac{s}{s^2 + \beta^2}.$$