okay we are going to work out the laplace transform of the constant function one and
this means F of T is equal to 140 and we were using the definition in this case for the see
the Laplace of 1 is going to be the integral from 0 to infinity and we always have e to the negative
T and we multiply this by by other this function is which is just a 1 right here and this is not
bad at all right and remember we have to integrate this with respect to T so that's put on PT right
here and now as you can see this is an improper integral so be sure you read it down the gentlemen
be we take the limit as n goes to infinity and of course you can use other letters right here up to
you but this book will use capital n anyways right here we will have the integral from 0 to
capital N and this is just e to the negative T DT all right and now we just have to do our integral
and then believe in our limit right so let me once again reduce down the limit as capital n
goes to infinity that's integrated well we have to integrate with respect to T that means s is
the constant e to the negative T we will first have e to the negative T and we look at this and
we differentiate is the derivative of negative T with respect to T is make the S going to indent
work with divided by that which is divided by negative s which is the same as 1 over negative s
here right and we have to integrate this from 0 to N and now plug in and into T plugging 0 into T and
we subtract two things so we will have the limit as n goes to infinity plugging into here we will
have 1 over negative e to the negative as times N and then this is the first part will subtract
so I can see 1/2 here we will have 1 over negative e to the negative s times T which is the zero and
that all we have right here okay so depending on how you want to do it and you see that this
is actually the troublesome because we have n goes infinity right here with this ad this is
just going to be nice because you have to zero so you know this is going to be nice but let's talk
about this well when your entry something either you get infinity or 0 right when you have power
being something like that imagine the whole thing right here be counting intimately now be bad isn't
it because infinity minus y over y over right here this is not going to be infinity the whole thing
will be infinity that's not good so we want this part we want this part to be equal to zero and how
can we have that well and is going towards path V infinity right here this is negative times s times
past infinity we have to make sure that s right here this right here has to be let me just put
it down we have to have so then you don't need s to be greater than zero that's a condition that we
had to set if n is greater than zero let's check this again negative times path D s times positive
this will give us negative infinity all together e to the negative infinity that will give us 0 times
1 over negative not be through so we want this to be 0 and we can do so by restrict ourselves
to have as d squared 0 that's all right so let me just show you you know we want it to be 0 if you
have this condition at the end so we'll put that down but they need to say this is going to be 0
and then minus minus da which becomes plus 1 over s 0 times s is 0 times negative which is still 0 e
to the 0 is just 1 so we have 1 over s right and then you see negative negative absolute one of
us anyway all in all is just going to be one over X and once again we have to attach that condition
that we put down earlier one over s so s is greater than zero and this will be great that's it