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# Part 2 of the transform of the sin(at)

## Video transcript

welcome back we were we were in the midst of figuring out the Laplace transform of sine of eighty when I was running out of time and so I just you know why this was this was the this is the definition of the Laplace transform of sine of eighty I said that also equals why this this is going to be useful for us since we're going to be doing integration by parts twice so I did it integration by parts once then integration by parts twice I said you know don't worry about the boundaries of the integral right now let's just worry about the indefinite integral and then after we solve for y let's just say Y is the indefinite version of this then we can evaluate the boundaries we got to this point and we made the realization after doing two integration by parts and being very careful not to hopefully make any careless mistakes we fit we realize wow this is our original Y right if I put the boundaries here that's that's the same thing as a Laplace transform of sine of eighty right that's our original Y so now and I'll switch colors just to avoid monotony this is equal to actually let me just this is why this is equal to Y right that was our original definition so let's add a squared over sine squared Y to both sides of this so this is equal to y plus I'm just adding this whole term to both sides of this equation plus a squared over s squared Y is equal to so this term is now gone so it's equal to this stuff and let's see if we can simplify this so let's see if we can let's factor out an e to the minus st actually let's factor out a negative e to the minus st so it's minus e to the minus st times time's side well sign of actually let's factor out it well let me just write one over s sign of eighty minus one over s squared cosine of eighty I really hope I haven't made any careless mistakes and so this we can add the coefficient so we get one plus a squared over s squared times y but we could that's the same thing as s squared over s squared plus a squared over s squared so it's s squared plus a squared over s squared Y is equal to e minus e to the minus s T times this whole thing sine of eighty minus one over s squared cosine of eighty and now this right here this is since we're dealing everything with respect to DT this is just a constant right so we could say a constant times the antiderivative is equal to this this is as good a time as any to evaluate the the boundaries right if this had a T here I would have to somehow get them get them back on the other side so that because the T's are involved in evaluating the boundaries since we're doing our our definite integral or improper integral so let's evaluate the boundaries now and we could have kept them along with us the whole time right and just factor it out the this this term right here but anyway so let's evaluate this from 0 to infinity and this should simplify things so the left hand side of this the right hand side of this equation when I evaluated at infinity what is e to the minus infinity well that is that is 0 we've established that multiple times and now it approaches 0 from the negative side they're still going to be 0 or it approaches 0 and then that times what will sign of infinity well sign just keeps oscillating right between negative 1 and plus 1 and so does cosine right so this is bounded so this thing is going to overpower these and if you're curious you can graph it this kind of forms an envelope around these oscillations so the limit as this approaches infinity is going to be equal to 0-0 and that makes sense right these are bounded between 0 and negative 1 of these this approaches 0 very quickly so 0 times something bounded between 1 and negative 1 another way to view it is the largest value this could equal is or the is is 1 times whatever coefficients on it and then this is going to 0 so it's like 0 times 1 anyway I don't want to focus too much on that you can play around with that if you like - this whole thing evaluated at 0 so - what's e to the minus 0 well that is e to the minus 0 is 1 right that's e to the 0 we have a minus 1 so it becomes plus 1 times now sine of 0 is 0 minus 1 over s squared cosine of 0 so what is what is 1 over cosine let's see 1 a cosine of 0 is 1 so we have minus 1 over s squared minus 1 over s squared times 1 so that is equal to that is equal to minus 1 over s squared and I think I made a mistake because I shouldn't be having a negative number here so let's let's let's backtrack and let's see where that mistake might maybe this isn't a negative number let's see infinity right this whole thing is 0 - let's see when when you put 0 here this becomes a minus 1 yeah I'm getting a let me see where am I so either this is a plus or this is a plus let's see where I made my mistake e to the minus s T oh I see where my mistake is right up here where I factored out an e - a - e to the minus s T right fair enough so that makes this 1 over s sine of 80 but I 5 factor out a minus e to the minus s T this becomes a plus right it was - here but I'm factoring out a minus e to the minus st so that's a plus this is a plus boy I'm glad that was not too difficult to find so then this becomes a plus and then this becomes plus thank God it would have been sad if I wasted two videos and I ended up with a careless negative number anyway so now we have s squared plus a squared over s squared times y is equal to this multiplied both sides times s squared over s squared plus a squared divide both sides by this and we get Y is equal to 1 over s squared 1 over s squared and actually let me make sure that that is right it's 1 over s squared right y is equal to 1 over s squared times s squared over s squared plus a squared and then these cancel out and let me make sure that I haven't made another careless mistake cuz I'm a feeling I have I have a feeling I have yep there I see the careless mistake and it was all in this term it was all in this term and I hope you don't mind my careless mistakes but I want you to see that you know I'm doing these things in real time and and I'm human in case you haven't realized already anyway so this I made the same careless mistake so i factor an e to the minus st here so it's plus but it was a over s squared so this is an a that's an a and so this is an A and so this is an a and so this is an a right this was an A and so we're left with and this is the correct answer a over s squared plus a squared so I hope that those careless mistakes didn't throw you off too much you know this these things happen when you do integration by parts twice with a bunch of variables but anyway now we are ready to add a significant entry into our table of Laplace transforms and that is that the Laplace transform I had an extra curl there that was unnecessary let me do it again the Laplace transform of sine of 80 is equal to a over s squared plus a squared and that's a significant entry and maybe a good exercise for you just to see how fun it is to do these integration by parts problems twice is to figure out the Laplace transform of cosine of eighty and I'll give you a hint it's s over s squared over over s squared plus a squared and it's nice that there's that symmetry there anyway I'm almost at my time limit and I'm I'm very tired working on this video so I'll leave it there and I'll see you in the next one