91Ó°ÊÓ

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{rr} 0, & 0 \leq t<1 \\ t^{2}, & t \geq 1 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<1 \\ t^{2}, & t \geq 1 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<3 \pi / 2 \\ \sin t, & t \geq 3 \pi / 2 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<3 \pi / 2 \\ \sin t, & t \geq 3 \pi / 2 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t<2 \\ 0, & t \geq 2 \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{rr} t, & 0 \leq t<2 \\ 0, & t \geq 2 \end{array}\right. $$

In Problems, write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} \sin t, & 0 \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. $$

Write each function in terms of unit step functions. Find the Laplace transform of the given function. $$ f(t)=\left\\{\begin{array}{lr} \sin t, & 0 \leq t<2 \pi \\ 0, & t \geq 2 \pi \end{array}\right. $$

Show how to use the Laplace transform to find the numerical value of the improper integral \(\int^{\infty} t e^{-2 t} \sin 4 t d t\).

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+y=f(t), \quad y(0)=0, \text { where } f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t<1 \\ 5, & t \geq 1 \end{array}\right. $$