yup yup ignore the poster and destroy his thread.
The problem is that 2 are expert and one is a moderator.
we're respecting you, and i gave you an alternate way to solve your problem which Drexel has completed giving you a full solution, absolutely, understandable, so if you don't get it, the only easy thing you need to do, is to ask, and that's all.
i actually complemented why i "broke" the integral into a sum and why using that bounds, so if still having any questions, ask, since at least my method it's pretty easy to get if you fully understand every content, which is not hard to digest!!!
for $\displaystyle \displaystyle \matcal{L} \{|\cos t|\} $ would you write it as $\displaystyle \displaystyle \int_{0}^{\frac{\pi}{2}} e^{-st} |\cos t| \ dt + \sum_{k=1}^{\infty}\int_{\frac{\pi (2k-1)}{2}}^{\frac{\pi (2k+1)}{2}} e^{-st} |\cos t | \ dt $ ?