Just noticed, that I have to integrate back by frequency dw.

Then formula should be:

But I don't know, how to integrate this and also w0 befrore depended by time also.

It seems to me, that you obviously know about this stuff.. so I can believe for you, that after integration I should get back s(t) as it was at the beginng.

I had doubts about sinus and delta function, but actually you explained it in post #10.

If we are speaking about requirement for the integral to converge, I wasn't convinced about this method, because I'm not sure wheather this 'infinity' criterie is acceptable or not when we got sinus.

As I understand, you are sure, that it is correct answer?