# Integration by parts homework help

Dustinsfl
$$\int_0^{infinity} \ e^{-s*t}*t*cos(t) dt$$

I tried integration by parts with u=t*cost and dv=e^(-s*t) but that didn't get anywhere.

I then tried: $$\L{t^n*g(t)}=(-1)^n d/ds[\int_0^{infinity} \ e^{-s*t}*cos(t) dt$$ but again nothing was working.

This is a Laplace Transformation where ft=t cos(t)

Homework Helper
Gold Member

I then tried: $$\L{t^n*g(t)}=(-1)^n d/ds[\int_0^{infinity} \ e^{-s*t}*cos(t) dt$$ but again nothing was working.

Do you mean that you tried applying the rule

$$\mathcal{L}\{t^n g(t)\}=(-1)^n\frac{d^n}{ds^n}\mathcal{L}\{g(t)\}$$

If so, that should be fairly easy, provided that you know what the Laplace transform of $\cos(t)$ is.

Dustinsfl

The problem was I get cos(t)/(s*e^(-st)) which is in determinant and using l'hopitals rule didn't help.

Also, the integral after separation by parts is now e time sin then you get an in determinant form with sin.

Homework Helper
Gold Member

The problem was I get cos(t)/(s*e^(-st)) which is in determinant and using l'hopitals rule didn't help.

How on Earth are you getting something that depends on $t$ after integrating over $t$?

Show your work so I can see where you are going wrong.

Dustinsfl

Work

Both methods attempted

Never mind the image won't load.