Posted by **Mike** on Sunday, May 3, 2009 at 5:05pm.

What is the laplace transform of (t^2)sin(t)?

- Math -
**Count Iblis**, Sunday, May 3, 2009 at 5:31pm
f(s) = integral from t = 0 to infinity of t^2 sin(t) exp(-s t) dt

To compute this integral consider first the Laplace transform of exp(i t). Using the fact that the integral from zeo to infinity of exp(-alpha t) is 1/alpha, you find that this is:

1/(s - i)

The Laplace transform of sin(t) then follows by taking the imaginary part of this:

Im[1/s-i] =Im[(s+i)/(s^2+1)] = 1/(s^2+1)

If you differentiate this twice w.r.t. s, you bring down a factor of t^2 in the integrand of the Laplace integral. So, the Laplace thransform is given by:

8 s^2/(s^2+1)^3 - 2/(s^2+1)

- Math -
**Count Iblis**, Sunday, May 3, 2009 at 6:19pm
Typo in last formula: The Laplace transform is:

8 s^2/(s^2+1)^3 - 2/(s^2+1)^2

## Answer this Question

## Related Questions

- calc 2 - (last one) ... Let f(t) be a function defined for all values of t. The...
- Calc2 - Let f(t) be a function defined for all values of t. The Laplace ...
- advanced math - what is the Laplace transform of e^4t(cos 2t) ?
- college math - the Laplace transform of e^4t(cos 2t)?? please i need help. ...
- ODE - what is the laplace transform of 5/2.e^t-3/2.e^-t
- Calc 3 - What is the laplace transform of: f(t) = 0, 0<t<2 t, 2<t ...
- Math - Using the table Laplace transform, determine the function f (t) of the ...
- Calculus - Find the Laplace transform of: f(t)=t-2e^(3t)
- Calculus - Find the inverse Laplace transform of: F(s)=3/(s-4)
- Calculus-HELP!!! - Find the inverse Laplace transform of: F(s)=2/s^4