Question

Definition 7.1.1 laplace transform let f be a function defined for t ≥ 0. then the integral {f(t)} = ∞ e−stf(t) dt 0 is said to be the laplace transform of f, provided that the integral converges. to find {f(t)}. f(t) = cos t, 0 ≤ t < π 0, t ≥ π

Answer 1

`\mathcal L\{f(t)\}=\displaystyle\int_(t=0)^(t\to\infty)f(t)e^(-st)\,\mathrm dt`

Given that


`f(t)=\begin{cases}\cos t&\text{for }0\le t<\pi\\0&\text{for }t\ge\pi\end{cases}`

the Laplace transform of
`f(t)` is given by the definite integral


`\displaystyle\int_(t=0)^(t\to\infty)f(t)e^(-st)\,\mathrm dt=\int_(t=0)^(t=\pi)\cos t\,e^(-st)\,\mathrm dt+\int_(t=\pi)^(t\to\infty)0\,\mathrm dt`

`=\displaystyle\int_0^\pi\cos t\,e^(-st)\,\mathrm dt`

`=((1-e^(-\pi s))s)/(s^2+1)`

(which you can find by integrating by parts twice)