## Question 1

*Determine whether the integrals are convergent or divergent. Evaluate those that are convergent.*

**This exercise is more to get us used to evaluating infinite integrals — none of these bar (iii) are necessarily related to Laplace transforms.**

*(i)*

#### Solution

We first try and evaluate

.

Try the substitution :

.

.

Now evaluating between the limits and :

.

Now taking the limit as :

.

The integral is convergent with value .

### (ii)

*We will do this by evaluating the indefinite integral, then put in the limits ( and ), then then take limit as .*

Consider

.

Note the function-derivative pattern – (or use LIATE), and hence let :

.

Now putting in the limits:

.

Now taking the limit as :

;

i.e. the integral is convergent with value .

### (iii)

Consider

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Now using a unit circle, the tables, a calculator, or otherwise, note that . That is the integral evaluates to:

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Now taking the limit as :

…

however this limit does not exist. does not get closer and closer to a particular value as increases — instead it oscillates between and . (Nice Wolfram Alpha graph http://www.wolframalpha.com/input/?i=Plot%5B1-Cos%5Bx%5D%2C{x%2C0%2C100}%5D).

## Question 2

*Find the Laplace transform of the zero function .*

### Solution

Using the definition:

.

## Question 3

*Find the Laplace transform of the following function :*

**I am under the impression that this would be a more Math 3.1 exercise — nothing like it will be on our exams.**

### Solution

Again, using the definition:

.

Hence splitting up the interval over which we integrate;

.

Clearly taking the limit as does not affect this so this is the answer.

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