Laplace transform of cos t and polynomials

in the last video we showed that the Laplace transform of F prime of T is equal to s times the Laplace transform of our function f minus f of zero now what we're going to do here is actually use this property that we showed is true and use it to fill in some more of the entries in our in our Laplace transform table that you'll probably have to memorize of sooner or later if you use Laplace transforms a lot but we already learned that the Laplace transform we know that the Laplace transform of the sine of a T is equal and we did a very hairy integration by parts problems to show that that is equal to a over s squared plus a squared so let's use this let's use these two things we know to figure out what the Laplace transform of cosine of a T is so the Laplace transform the Laplace transform of cosine of a T is equal to what well if we assign assume that Laplace transform of cosine of a T is the derivative of some function what what is it the derivative of right if I were to let me do it on the side if F prime of T is equal to cosine of a T what is a potential f of T what is a potential f of T well it's the antiderivative and we can just forget about the constant because we just have to know NF of T for which this is true so what's the antiderivative cosine of it what's 1 over a it's 1 over a sine of 80 right so if this is if this is f prime of T then that is equal to s times that's equal to s times the Laplace transform of its of its antiderivative or 1 over a sine of 80 minus the drew antiderivative evaluated at 0 minus 1 over a sign of well a times 0 is 0 well sine of 0 is 0 so this whole term goes away so this is equal to well this is a constant right this 1 over a and we showed that the Laplace transform is a linear operator so we can take it out so this is equal to s over a times the Laplace transform of sine of a T and that is equal to s over a times a over s squared plus a squared and the a is canceled out and that was much simpler than the integration by parts we have to do to figure this out so then we get that the Laplace transform the Laplace transform of cosine of a T is equal to s over s squared plus a squared and in 3 minutes we filled in another table in our Laplace transform table and this is and now we have you know the two most important trig function let's keep going we haven't really done much much with polynomials we know a couple of things we know that the whoops we know that the Laplace transform we did this already we know that the Laplace transform of 1 is equal to 1 over s so let's see if we could use this and the fact that the Laplace transform of F prime is equal to s times the Laplace transform of F - f of zero or another way let's let's rewrite let's rearrange this like if we know F how can we figure out a salats transforms in terms of F Prime and F of zero so let's let's just rearrange this equation so we get the Laplace transform of F Prime I could write of T but that gets monotonous plus F of zero is equal to s times the Laplace transform of F divide both sides by ass let me put the Laplace transform of an and I'm also going to switch to the side so I get the Laplace transform my elves are getting funky the Laplace transform of F is equal to 1 over s I'm just dividing both sides by s 1 over s times this times the Laplace transform of my derivative plus my function evaluated at 0 and let's see if we can use this to figure out this and this to figure out some some more useful Laplace transforms well what is the Laplace transform of the Laplace transform what is the Laplace transform of F of T is equal to T the Laplace transform of F of T is equal to T well just let's use this property this is going to be equal to 1 over s times the Laplace transform of the derivative well what's the derivative of T the derivative of T is 1 so it's a little plus transform of 1 minus F of 0 when T equals 0 this becomes 0 minus 0 so the Laplace transform of T is equal to 1 over s times the Laplace transform of 1 well that's just 1 over s so it's 1 over s squared minus 0 interesting Laplace transform of 1 is 1 over s Laplace transform of T is 1 over s squared let's figure out what the Laplace transform of T squared is and I'll do this one in green and maybe we'll see a pattern emerge the Laplace transform of T squared well it equals one over s times the Laplace transform of of its derivative so what's the Laplace transform of 2t plus this evaluated zero well that's just zero so this is equal to well we could just take this constant out this is equal to 2 over s times the Laplace transform of T well what does that equal that is equal to we just solved in one of our s squared so it's 2 over s times 1 over s squared so it's equal to 2 over s to the third 2 over s to the third fascinating well let me ask you let me ask you a well let me just do T to the third and I think then you'll see you'll see the pattern the pattern will emerge the Laplace transform and I this is actually kind of fun I recommend you do it it's it's somehow satisfying it's much more satisfying than integration by parts so the Posche transform of 2/3 is 1 over s times the Laplace transform of its derivative which is 3t squared which is take the constant out because it's a linear operator 3 over s times the Laplace transform of T squared so it equals what's the Laplace transform of T square is 2 over s 1/3 so this equals 3 times 2 over what s to the fourth and you could put T over N here and use an inductive argument to figure out a general formula and that general formula is and I think you see the pattern here whatever my exponent is the Laplace transform has an S in the denominator with one larger exponent and then the numerator is the factorial of my exponent so in general and this is one more entry in our Laplace transform table the Laplace transform of T to the nth power is equal to n factorial over s to the n plus 1 s to the n plus 1 that's a parentheses I guess I didn't have to write those parentheses it just confuses it but anyway this sometimes looks like a fairly you know when you see this in a Laplace transform table it seems intimidating oh boy have ends and I have n factorials and all that but it's just saying with this pattern we show whatever you know T to the third increase it by one so s to the fourth put in that the denominator and take 3 factorial on the numerator which is 6 right and that's all it is so we have in using the property the derivative property of Laplace transform we figured out the Laplace transform of of cosine of a T and the Laplace transform of any well really any polynomial right because it's a linear operator so now we can you know we know T to the nth power T to the whatever power and multiply it by constant so we know the basic trig functions we know the exponential function and we know we know how to take the Laplace transform of polynomials see you in the next video