# Thread: laplace transform with powers of t*exponential

1. ## laplace transform with powers of t*exponential

Hi all,
I have a question.

f(t) =t^3*e^-4t using theorm 1 and 2.

what im wondering is can I take the t^3 outside the integral as a constant

t^3| e^-st * e^-4t (between 0 and infinity)

then

t^3 [ e-st/-s + e^-4t/s] (between 0 and infinity) or should I be thinking of partial fractions all of which are kind of new to me.

2. ## Re: laplace transform with powers of t*exponential

by the way I need to show all steps rather than refering to the tables, according to which I should end up with

6/s^4 * 4/s+4

3. ## Re: laplace transform with powers of t*exponential

Originally Posted by tolka
Hi all,
I have a question.

f(t) =t^3*e^-4t using theorm 1 and 2.

what im wondering is can I take the t^3 outside the integral as a constant
No, you can't, because it is NOT a constant. The Lalace transform is, by definition
$\int_0^\infty f(t)e^{-st}dt= \int_0^\infty t^3e^{-4t}e^{-st}dt= \int_0^\infty t^3e^{(-4-s)t}dt$
Do that "by parts" three times.

t^3| e^-st * e^-4t (between 0 and infinity)

then

t^3 [ e-st/-s + e^-4t/s] (between 0 and infinity) or should I be thinking of partial fractions all of which are kind of new to me.

4. ## Re: laplace transform with powers of t*exponential

Do you mean partial fractions or int by parts?