I suck horribly at using limits and absolute value signs to manipulate integrals but here is my attempt at the attached question. Im pretty close to the answer... I imagine I'm doing something wrong in the absolute value part...

While I was typing this I may have solved it but anyway here goes...

a)

.

b)

Now here's where my 'say it to get the right answer' logic comes in.

Since f is of moderate decrease (or perhaps due to the inequality on

...) then we can find a h such that the first integral on the RHS is zero yes?

So assuming the above is true...

We apply absolute values to get...

(Can I do that? Take the absolute value inside, thus getting rid of the exponentials..?)

At this point onwards I'm just working things out as I go along...

(?)

.

So...

Where do I go wrong (I say that since I have a feeling something is not quite right despite getting the right answer!)