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### Course: Differential equations>Unit 3

Lesson 1: Laplace transform

# L{sin(at)} - transform of sin(at)

Laplace Transform of sin(at) (part 1). Created by Sal Khan.

## Want to join the conversation?

• @ shouldn't [(-1/s) e^(-st) sin(at)] also be evaluated from zero to infinity ?
• That's true. Later at Sal notices it
• hey by the way, this "hairy problem" becomes a smoothly shaved, easy peasy integral if we use complex substitution (Euler's formula) to sub in for Sin at Or Cos at. It comes down to (s+ai)/(s^2+a^2). And so if we are wanting the
L (sin at), we note that the Sin is the imaginary part of the Euler formula, so we choose the imaginary part of the top... L (sin at) = a/(s^2+a^2)! Super easy. And we can use that same answer above for L (cos at). Since cos is the Real part of the Euler formula then its the Real part of the solution... Therefore,
L(cos at)= s/(s^2+a^2) ! Cool stuff. and MUCH easier integration to get these!
• I have a way to understand the integration by parts graphically:
https://rubenayla.blogspot.com/p/visualizar-integrales-por-partes.html

Imagine a rectangle of sides u and v. Then the area of the rectangle would be u*v. Imagine an awkward function inside making a strange area that you can integrate u dv. That area would be the area of the entire rectangle minus the area of the other side, which graphically turns to be the integral of v du
• I heard that sin, cos, and tan stood for sine cosine tangarine. Is this true?
• Yes...furthermore csc = cosecant, sec = secant, cot = cotangent.
• sir in laplace what is (D2+1)y=tcost with y(0)=0, y`(0)=0 sir pleases tell how to solve
• Wouldn't this have been easier using Euler's formula, rewriting sin(ax) as 1/(2i)*(e^(i*ax) - e^(-i*ax)), converting the problem into the integration of two simple exponentials? I guess I'm not sure why parts is better...
• Wouldn't it be easier to integrate if we took u_prime = sin(at) and v = e^(-st). We can finish integrating in one step by this process. But the solution looks a bit compact as well. What is the significance of integrating in the way it is done in the video?
(1 vote)
• There is no significance in integrating either way. They both take approximately the same amount of work, and neither have any advantages over the other.