# Determine the Laplace transform of the given function f. f(t)=(t -1)^2 u_2(t)

Question
Laplace transform
Determine the Laplace transform of the given function f.
$$f(t)=(t -1)^2 u_2(t)$$

2021-03-08
Step 1
Here laplace transform of given function is to be calculated.
Given function is $$f(t)=(t -1)^2 u_2(t)$$
So expand the function as follows,
$$f(t)=t^2u(t)+u(t)-2tu(t)$$
Take Laplace of the function f(t)
Step 2
Then,
$$L(f(t))=L(t^2u(t)+u(t)-2tu(t))$$
$$=\frac{2}{s^3}+\frac{1}{s}-\frac{2}{s^2}$$
Hence the Laplace Transform of the $$f(t)=(t-1)^2u(t) \text{ is } \frac{2}{s^3}+\frac{1}{s}-\frac{2}{s^2}$$

### Relevant Questions

determine a function f(t) that has the given Laplace transform
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$$F(s)=\int_0^\infty e^{-st} f(t)dt \(\text{where we assume s is a positive real number. For example, to find the Laplace transform of } f(t)=e^{-t} \text{ , the following improper integral is evaluated using integration by parts:} \(F(s)=\int_0^\infty e^{-st}e^{-t}dt=\int_0^\infty e^{-(s+1)t}dt=\frac{1}{s+1}$$
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True or False

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$$\displaystyle f{{\left({t}\right)}}={\left\lbrace\begin{matrix}{1}-{t}&{0}<{t}<{1}\\{0}&{1}<{t}\end{matrix}\right.}$$
Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
$$F(s)=\int_0^\infty e^{-st}f(t)dt$$
where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^{-t}, the following improper integral is evaluated using integration by parts:
$$F(s)=\int_0^\infty e^{-st}e^{-t}dt=\int_0^\infty e^{-(s+1)t}dt=\frac{1}{(s+1)}$$
Verify the following Laplace transforms, where u is a real number.
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$$F(s) =\int_0^\infty e^{-st}f(t)dt$$