Explanations Of These Laplace Transforms and Properties

Hello, all.

I was on this site previously, but, forgot my username and etc. I would like to say the website looks like it had a great improvement to it, so good job to those who worked on it. I have a few questions about these Laplace transforms.

Some Information: I have taken a Differential Equations previous semester to this course I am taking now which calls for to use this, which is a Circuits class. We never really covered Laplace Transforms too in depth just a basic here what they are and some stuff to remember whilst using a table. I am not one too big on tables until I know the work in between unless it calls for some massive learning I cannot comprehend yet on my current level of mathematical knowledge. So in all these current Laplace Transforms and properties, and unit functions i.e step, impulse, pulse, ramp functions are being used in conjunction of my circuits analysis course if this helps anyone.

**Property/Transform 1: **

$\displaystyle \int_{0}^t f(\lambda) d\lambda = \frac{1}{s} F(s) $

I was told that lambda was just a dummy variable on integration. But, how would you so to speak integrate or arrive and the result of this property? Because I was given a simple problem to evaluate this unit step function...please do not solve, I want to learn this.

$\displaystyle \int_{1}^t \mu(\lambda) d\lambda $

I wouldn't assume the professor wanted us to evaluate simply by saying using property so and so on table so and so gives yada yada. That is useless to me at least. I will post more as we progress here so thanks to anyone who takes the time to answer.