Question

laplace inverse of 1/2(e^(-2x))((x^2-2x-1)/(x^3+x-2))

Answer 1

The laplace transform or the inverse transform?

Answer 2

the inverse laplace transform where t is greater than or equal to 2

Answer 3

So they want you to find the transform and then calculate the transform that will get you back? Because that is not a function of t...?

Answer 4

here is a print screen of the problem in the attachment. i honestly don't know what i'm doing! hope you can help!

Answer 5

I can help! Find the laplace transform: The laplace transform is defined as:
\[\int\limits_0^{\infty}f(t)e^{-st}dt\]
Do you know how to evaluate integrals of this kind?

Answer 6

do i separate the f(t) and e^(-nt)? and then take the limit of each as n goes to infinite?

Answer 7

So this is what I would do, notice with t < 1 the function is zero, so your integral because:
\[\int\limits_1^{\infty}(t^2-2t+2)e^{-st}dt\]

Answer 8

becomes**

Answer 9

but i thought at t=0 the function would equal 2...

Answer 10

So evaluate each piece individually:
\[\int\limits_1^{\infty}t^2e^{-st} - 2 \int\limits_1^{\infty} t e^{-st} dt +2 \int\limits_1^{\infty}e^{-st}dt\]
Use the definition given.

Answer 11

You can arbitrarily define a function to be anything at any given point, they are just declaring it to be that when t<1 /shrug

Answer 12

Ready to work out this integral?

Answer 13

yes

Answer 14

i thought the integral would be infinite...

Answer 15

especially since e^x does not converge

Answer 16

Its not e^x, its e^-st, the negative makes a huge difference. Okay, on the first integral use integration by parts:
\[u=t^2, du=2t dt; dv = e^{-st}dt, v = -e^{-st}/s \implies \int\limits_1^{\infty}t^2e^{-st} \]\[= -t^2e^{-st}/2|_1^{\infty}+ 2 \int\limits_1^{\infty}te^{-st}dt\]

Answer 17

Those should be over s actually.

Answer 18

Perform integration by parts again and you're done. Let a be a finite value, evaluate the integrals and take the limits as a goes to infinity and you'll be done.

Answer 19

so the function after solving the integral will be the answer to the inverse laplace?

Answer 20

is s a variable or a constant. is t the only variable?