# Section 6.1 – Definition of the Laplace Transform

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Use the definition to find the Laplace transforms of
1. $$f(t)=e^{at}$$ where $$a$$ is a non zero real number.
2. $$f(t)=\sin (bt)$$ where $$b$$ is a non zero real number.
3. $$\dfrac{b}{s^2+b^2}$$ provided $$s>0$$
4. $$\displaystyle f(t)=\left\{\begin{array}{ll}5-t\quad&0\leqslant t<2\\ 3t&2\leqslant t.\end{array}\right.$$
5. $$\dfrac{1}{s^2}$$ provided $$s>0$$

1. $$\dfrac{1}{s-a}$$ provided $$s>a$$
2. $$\dfrac{b}{s^2+b^2}$$ provided $$s>0$$
3. $$\dfrac{5}{s}-\dfrac{1}{s^2}+e^{-2s}\left(\dfrac{3}{s}+\dfrac{4}{s^2}\right)$$ provided $$s>0$$
4. $$\dfrac{1}{s^2}$$ provided $$s>0$$
5. $$\dfrac{2}{s^3}$$ provided $$s>0$$

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2. Find the inverse Laplace transform of the following functions.
1. $$F(s)=\dfrac{4}{(s-2)^5}$$
2. $$F(s)=\dfrac{8s^2-4s+12}{s(s^2+4)}$$
3. $$F(s)=\dfrac{2s-3}{s^2+2s+10}$$

1. $$f(t)=\dfrac{e^{2t}\cdot t^4}{6}$$
2. $$f(t)=3+5\cos(2t)-2\sin(2t)$$
3. $$f(t)=2e^{-t}\cos(3t)-\dfrac{5}{3}e^{-t}\sin(3t)$$
Video Errata: In part (c), at the 20:29 mark, there is a sign error. The line should say $$2\cdot \dfrac{s+1}{(s+1)^2+3^2}-\dfrac{5}{3}\cdot\dfrac{3}{(s+1)^2+3^2}$$. We then have that $$f(t)=2e^{-t}\cos(3t)-\dfrac{5}{3}e^{-t}\sin(3t).$$

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