Proceed as in example 5 and find the laplace transform of f...

Proceed as in Example 5 and find the Laplace transform of f * g using the Convolution Theorem. Do not evaluate the convolution integral before transforming. (Write your answer as a function of s.) $\mathcal{L}\left\{t \int_{0}^{t} \sin(\tau) d\tau\right\}$ $\frac{1}{s(s^2+1)}$

Transcript

00:02 Okay, so let's do this.

00:05 So what is the loplas? so first of all, let's write this as a convolution.

00:11 Cosine of tau, d tau.

00:14 We know that this has the form of the convolution of the cosine function convolved with the one function.

00:25 Because convolutions always have the form of, because this is the same thing as integral.

00:33 From 0 to t, cosine of tau, 1 of, well, 1 of tau minus t.

00:53 Or actually, you probably want t minus tau, d tau.

01:02 So this is the same thing.

01:03 So what this tells you is that you can write this as the cosine function convoyed with one function.

01:09 And we know that the laplace transform of a convolution of f composed with g, is just the laplace transforms is just a multiplication of the laplace transform so i guess okay let me write this in a different way this is the laplace transform of f times the laplace transform of g okay so all this tells us is to find the laplace transform of the cosine function and one function okay so let's do this what's the laplace transform of one well this is integral from 0 to infinity of e to the minus st d t okay and you can just do just do a u substitution right so take u equals st then you get this equal to 1 over s integral from still from 0 to infinity of e to the minus u d u okay and if you integrate this this just becomes 1 over s minus e to the minus u from 0 to infinity which is just equal to 1 over s because e to the minus infinity is equal to 0 and e to the 0 is 1.

02:39 So this is just 1 over s.

02:41 Now what about the cosine function? so this is going to be 0 to infinity of e to the minus s t.

02:58 Okay, the easiest way to do this is to do integration by parts twice...

Question

Proceed as in Example 5 and find the Laplace transform of f * g using the Convolution Theorem. Do not evaluate the convolution integral before transforming. (Write your answer as a function of s.) $\mathcal{L}\left\{t \int_{0}^{t} \sin(\tau) d\tau\right\}$ $\frac{1}{s(s^2+1)}$

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