Question

If

F(s) = script L{f(t)}

and

n = 1, 2, 3, . . . ,

thenscript L{tnf(t)} = (−1)n

dn
dsn
F(s).

Evaluate the given Laplace transform. (Write your answer as a function of s.)

script L{t cos 5t}

Answer 1

`(s^2-25)/((s^2+25)^2)`

Explanation:

Let's use the definition of the Laplace transform and the identity given:
`\mathcal{L}[t \cos 5t]=(-1)F'(s)` with
`F(s)=\mathcal{L}[\cos 5t]`.

Now,
`F(s)=\int_0 ^(+ \infty)e^(-st)\cos(5t) dt`. Using integration by parts with u=e^(-st) and dv=cos(5t), we obtain that
`F(s)=(1)/(5)\sin(5t)e^(-st) |_(0)^(+\infty)+(s)/(5)\int_0 ^(+ \infty)e^(-st)\sin(5t) dt=\int_0 ^(+ \infty)e^(-st)\sin(5t) dt`.

Using integration by parts again with u=e^(-st) and dv=sin(5t), we obtain that

`F(s)=(s)/(5)((-1)/(5)\cos(5t)e^(-st) |_(0)^(+\infty)-(s)/(5)\int_0 ^(+ \infty)e^(-st)\sin(5t) dt)=(s)/(5)((1)/(5)-(s)/(5)\int_0^(+ \infty)e^(-st)\sin(5t) dt)=(s)/(5)-(s^2)/(25)F(s)`.

Solving for F(s) on the last equation,
`F(s)=(s)/(s^2+25)`, then the Laplace transform we were searching is
`-F'(s)=(s^2-25)/((s^2+25)^2)`