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

$\displaystyle\int_{t=0}^{t\to\infty}f(t)e^{-st}\,\mathrm dt=\int_{t=0}^{t=\pi}\cos t\,e^{-st}\,\mathrm dt+\int_{t=\pi}^{t\to\infty}0\,\mathrm dt$
$=\displaystyle\int_0^\pi\cos t\,e^{-st}\,\mathrm dt$
$=\dfrac{(1-e^{-\pi s})s}{s^2+1}$

(which you can find by integrating by parts twice)