## Question 3

*Find, from first principles, *.

### Solution

Using the definition:

To ease the steps, first we will evaluate the indefinite integral:

.

This integral needs to be integrated by parts (*this hint would be proffered in an exam situation*). Using the LIATE rule, choose and . Now

, and

.

In this context is a constant and it will be handy to have it out the front so it can be easily taken out of an integral (.) Hence using the integration by parts formula:

.

Now this second integral will also need integration by parts. By LIATE choose and . Hence

, and

.

Using the integration by parts formula:

.

Therefore,

.

Now putting in the limits to :

Now three algebraic facts:

- For any real (or complex) number ,

This yields:

.

Now we have an analysis fact:

Suppose . Then grows much faster than any so that

(*Alternatively if we demonstrate that we know what we are doing we can simply state , and tend to zero*). Taking the limit as :

.

In other words,

,

as we would expect, looking at the tables.

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