- [Instructor] We want to find the Laplace transform,
which is big F of s of the function f of t equals three.
Look to the definition below.
Suppose that f of t is a piecewise continuous function
for t greater than or equal to zero.
The Laplace transform of f of t is equal to
the integral from zero to infinity,
of f of t times e raised to the power of negative s t d t,
where s is greater than zero.
So for this example f of s is equal to the Laplace transform
of three which equals the integral from zero to infinity
of three times e raised to the power of negative s t d t.
Now because we have an improper integral,
we need to re-write this using limit notation.
So for a quick review,
looking at case one,
integral from a to infinity of f of x d x,
equals the limit as b approaches infinity,
of the integral from a to b of f of x d x.
So notice how we substitute b for infinity,
and then introduce this limit notation.
So this is equal to the limit as b approaches infinity,
of the integral from zero to b,
of three times e raised to the power of negative s t d t.
Now for our next step,
let's go ahead and find the anti-derivative of
three times e raised to the power of negative s t d t.
Then we'll come back and evaluate this def-integral,
and then find the limit.
So let's first find the integral of three times e,
raised to the power of negative s t d t.
Notice how we need to perform u substitution
where we'd have u equal negative s t and
differential u is equal to the derivative of
negative s t with respect to t times d t,
which would be negative s d t.
Dividing both sides by negative s.
We can write this as
negative one divided by s d u equals d t.
So in terms of u we can re-write this as the integral.
Again d t is equal to negative one divided by s.
So we can write this as negative three divided by s d u,
this would just be e to the u.
So the anti-derivative is just negative three
divided by s e to the u plus c.
But u is negative s t so we have
negative three divided by s e to the negative s t plus c.
But remember we had a def-integral.
So going back to our first slide.
We had the limit as b approaches infinity of,
again this is a def-integral,
so we can leave off the plus c,
and the anti-derivative is negative three divided by s,
e to the negative s t,
our limits of integration are
from t equals zero to t equals b.
So now for our next step we'll find
big F of b minus big F of a,
and then we'll determine this limit.
So when t equals b,
we'd have negative three divided by s times e,
raised to the power of negative s b,
minus when t is zero,
we'd have negative three divided by s times e,
raised to the power of negative s times zero.
Simplifying we have the limit as b approaches infinity of,
for this first term let's write this using a positive
exponent by moving e raised to the power of negative s b
down to the denominator.
So we'd have negative three divided by s times e raised
to the power of positive s b.
And then here either the zero is equal to one,
so we have minus negative three divided by s.
Which becomes plus three divided by s.
And now determining this limit,
remember s is greater than zero.
So as b approaches infinity,
the exponent on e approaches infinity and therefore
the exponent is increasing without bound.
And the numeral's fixed at negative three.
So this term then approaches zero.
This term is not affected by b.
And therefore the limit is equal to three divided by s,
which is the Laplace transform of f of t equals three.
So to summarize,
the Laplace transform of three equals big F of s,
which we now know equals three divided by s.
So going back to our first slide,
of course we entered three divided by s.
Now I do want to mention after we do several examples,
we'll stop using the limit definition
to determine Laplace transforms,
similar to how when we first found derivatives,
we used limit definitions,
but then we started using a table of derivatives.
And just like there's a table of derivatives,
there's also a table of Laplace transforms.
And here are a few formulas for basic Laplace transforms.
Looking at this first row,
if we have f of t equals one,
the Laplace transform is big F of s equals one divided by s.
Notice how because we had f of t equals three,
if we multiply one over s by three,
we get the Laplace transform of the constant function
f of t equals three.
So for each example,
after we use a limit definition
to find the Laplace transform,
we'll also look for the corresponding formula
in that Laplace transform table.
I hope you found this helpful.