Question

I need to find the Laplace Transform of f(t)= 0 if t<2, 3sin(pi*t) if 23.

Answer 1

As far as making the function in terms of step functions, is it like 3sin(pi.t).u(t-2) - 3sin(pi.t).u(t-3)?

I think this is what we learned in class.

Answer 2

Or just write it down. You want
\[ \int_2^3 e^{-st} \sin(\pi t) \ dt \]

Just evaluate that integral explicitly.

Answer 3

...because for your function f(t), the Laplace transform
\[ \int_0^{\infty} e^{-st} f(t) \ dt \]
is equal to that integral

Answer 4

write it as step function first

Answer 5

Alright, thanks, but I think it'd be easier to write it in terms of step functions.

Answer 6

exactly! thats what im doing, i'll test my answer now.

Answer 7

Are you kidding? The integral I wrote down is easy to evaluate.

Answer 8

but whatever you like.

Answer 9

Is that really correct tho because it is on an interval? the value of f(t) is only 3sin(pi.t) from 2<t<3. else it is 0.

Answer 10

Exactly. So the integrand is everywhere else equal to zero. Hence the once place where you are not integrating zero is on the interval [2,3]

Answer 11

You may find this identity useful:
\[ \int e^{ax} \sin(bx) dx = \frac{1}{a^2 + b^2} (a e^{ax} \sin(bx) - b e^{ax} \cos(bx) ) \]

Answer 12

Okay thanks. I am curious how I would do it with step functions tho. I'll ask another question :)