Hi everyone! Read through the material below and watch the videos, then give the WeBWorK assignment a try.

### Review: Finding the Laplace Transform of a function

You may wish to refer to this Table of Laplace Transforms during today’s lesson.

**Example 1.** Find the Laplace Transform of each function, and determine the interval on which it is defined.

- $t^{2}+4 t^{3}-7 t^{6}$
- $\sin t+\cos 3 t$
- $5 e^{2 t}-4 \sin 3 t$
- $5 t^{2}+3 \sin 5 t-2 e^{6 t} \cos 2 t$

### Find the inverse Laplace Transform

Now we consider the inverse problem – if I give you the Laplace Transform of a function, can you find the function it came from?

**Example 2.** Find the inverse Laplace Transform of each function.

- $\frac{3}{s}+\frac{4 !}{s^{5}}+\frac{1}{s-7}, \quad s>7$
- $\frac{1}{s^{2}+25}+\frac{s}{s^{2}+25}, \quad s>0$
- $\frac{1}{(s-4)^{2}+9}, \quad \mathrm{~s}>4$
- $\frac{1}{(s-6)^{7}}+\frac{5}{2 s-7}, s>6$
- $\frac{5}{s^{2}-8 s+41}, s>4$

That’s it for today! Try out the WeBWorK assignment, and let your professor know if you have questions.

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