Question

Laplace Transform f(t) = cos(t), 0 ≤ t < π 0, t ≥ π Complete the integral(s) that defines ℒ{f(t)}. ℒ{f(t)} = π dt + [infinity] π dt Find ℒ{f(t)}. (Write your answer as a function of s.) ℒ{f(t)} = (s > 0) Need Help?

Answer 1

Answer:

L[f(t)=s/(1+s^2)]

Step-by-step explanation:

The Laplace Transform is given by the integral:

$L[f(t)]=\int_0^\infty e^{-st}\ f(t)dt$

by replacing f(t)=cost we get

$\int_0^{\infty} e^{-st}costdt=[e^{-s(\infty)}sin(\infty)-1sin(0)]-s\int_{0}^{\infty}e^{-st}sintdt\\\\=0+s[-e^{-s(\infty)}cos(\infty)+e^{s(0)}cost(0)-s\int_0^{\infty}e^{-st}costdt]\\\\=0+s[0+1-s\int_0^{\infty}e^{-st}costdt]=s-s^2\int_0^{\infty}e^{-st}costdt\\\\(1+s^2)\int_0^{\infty}e^{-st}costdt=s\\\\\int_0^{\infty}e^{-st}costdt=\frac{s}{1+s^2}$

hope this helps!!