What is the half-range Fourier sine series for the function f(t) = t sin(t)?

[NotStine]

Homework Statement

Question: Find the half-range Fourier sine series for the function f(t) = t sin(t)

Problem: According to all the examples I have gone through, they all have a limit when asking for the half-range. However, my teacher, in the question posted above, has not specified any limits. Is this a typing error? If not, can you please nudge me in the right direction.

Homework Equations

The Attempt at a Solution

None yet. I'm under the impression that question may have been typed wrong.

 

[rootX]

I think -pi to pi are standard ...
Currently, it is an even function, I can suggest making it odd t*cos(t) and finding Fourier series from 0 to 2pi.

 

[NotStine]

Ok here is what I gather so far:

I am looking for the sine half-series, which is b_n.Sin(nt) from the Fourier series.

So,

b_n = I{t.sin(t).sin(nt)} between 0 and 2pi

... which goes to ...

b_n = I{t.sin_t(1+n)} between 0 and 2pi?

Is that correct?

EDIT: ... which gives me 0. I think I misunderstood.

Reading your suggestion again, you have changed t.sin(t) to t.cos(t)... Why is that? I can see we get an odd function (odd . even) but not sure how we came about the change...

Apologies in advance if I sound retarded, but 2 lectures on Fourier was no way near enough in my opinion.

 

[NotStine]

Any ideas?